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BEGIN:VEVENT
SUMMARY:Narad Rampersad (University of Winnipeg)
DTSTART:20200505T123000Z
DTEND:20200505T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/1
DESCRIPTION:Title: Ostrowski numeration and repetitions in words\nby Narad Rampersad
(University of Winnipeg) as part of One World Numeration seminar\n\n\nAbst
ract\nOne of the classical results in combinatorics on words is Dejean's T
heorem\, which specifies the smallest exponent of repetitions that are avo
idable on a given alphabet. One can ask if it is possible to determine th
is quantity (called the *repetition threshold*) for certain families of in
finite words. For example\, it is known that the repetition threshold for
Sturmian words is 2+phi\, and this value is reached by the Fibonacci word
. Recently\, this problem has been studied for *balanced words* (which ge
neralize Sturmian words) and *rich words*. The infinite words constructed
to resolve this problem can be defined in terms of the Ostrowski-numerati
on system for certain continued-fraction expansions. They can be viewed a
s *Ostrowski-automatic* sequences\, where we generalize the notion of *k-a
utomatic sequence* from the base-k numeration system to the Ostrowski nume
ration system.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Solomyak (University of Bar-Ilan)
DTSTART:20200519T123000Z
DTEND:20200519T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/2
DESCRIPTION:Title: On singular substitution Z-actions\nby Boris Solomyak (University
of Bar-Ilan) as part of One World Numeration seminar\n\n\nAbstract\nWe con
sider primitive aperiodic substitutions on $d$ letters and the spectral pr
operties of associated dynamical systems. In an earlier work we introduced
a spectral cocycle\, related to a kind of matrix Riesz product\, which ex
tends the (transpose) substitution matrix to the $d$-dimensional torus. Th
e asymptotic properties of this cocycle provide local information on the (
fractal) dimension of spectral measures. In the talk I will discuss a suff
icient condition for the singularity of the spectrum in terms of the top L
yapunov exponent of this cocycle. \n\nThis is a joint work with A. Bufetov
.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Carton (Université de Paris)
DTSTART:20200512T123000Z
DTEND:20200512T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/3
DESCRIPTION:Title: Preservation of normality by selection\nby Olivier Carton (Univers
ité de Paris) as part of One World Numeration seminar\n\n\nAbstract\nWe f
irst recall Agafonov's theorem which states that finite state selection pr
eserves normality. We also give two slight extensions of this result to no
n-oblivious selection and suffix selection. We also propose a similar stat
ement in the more general setting of shifts of finite type by defining sel
ections which are compatible with the shift.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Célia Cisternino (University of Liège)
DTSTART:20200526T123000Z
DTEND:20200526T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/4
DESCRIPTION:Title: Ergodic behavior of transformations associated with alternate base exp
ansions\nby Célia Cisternino (University of Liège) as part of One Wo
rld Numeration seminar\n\n\nAbstract\nWe consider a p-tuple of real number
s greater than 1\, $\\boldsymbol{\\beta} = (\\beta_1\,\\dots\,\\beta_p)$\,
called an alternate base\, to represent real numbers. Since these represe
ntations generalize the 𝛽-representation introduced by Rényi in 1958\,
a lot of questions arise. In this talk\, we will study the transformation
generating the alternate base expansions (greedy representations). First\
, we will compare the $\\boldsymbol{\\beta}$-expansion and the $(\\beta_1*
\\cdots*\\beta_p)$-expansion over a particular digit set and study the cas
es when the equality holds. Next\, we will talk about the existence of a m
easure equivalent to Lebesgue\, invariant for the transformation correspon
ding to the alternate base and also about the ergodicity of this transform
ation. \n\nThis is a joint work with Émilie Charlier and Karma Dajani.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Baker (University of Birmingham)
DTSTART:20200609T123000Z
DTEND:20200609T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/6
DESCRIPTION:Title: Equidistribution results for self-similar measures\nby Simon Baker
(University of Birmingham) as part of One World Numeration seminar\n\n\nA
bstract\nA well known theorem due to Koksma states that for Lebesgue almos
t every $x>1$ the sequence $(x^n)$ is uniformly distributed modulo one. In
this talk I will discuss an analogue of this statement that holds for fra
ctal measures. As a corollary of this result we show that if $C$ is equal
to the middle third Cantor set and $t\\geq 1$\, then almost every $x\\in C
+t$ is such that $(x^n)$ is uniformly distributed modulo one. Here almost
every is with respect to the natural measure on $C+t$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Matheus (Ecole Polytechnique)
DTSTART:20200616T123000Z
DTEND:20200616T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/7
DESCRIPTION:Title: Approximations of the Lagrange and Markov spectra\nby Carlos Mathe
us (Ecole Polytechnique) as part of One World Numeration seminar\n\n\nAbst
ract\nThe Lagrange and Markov spectra are closed subsets of the positive r
eal numbers defined in terms of diophantine approximations. Their topologi
cal structures are quite involved: they begin with an explicit discrete su
bset accumulating at $3$\, they end with a half-infinite ray of the form $
[4.52\\cdots\,\\infty)$\, and the portions between $3$ and $4.52\\cdots$ c
ontain complicated Cantor sets. In this talk\, we describe polynomial time
algorithms to approximate (in Hausdorff topology) these spectra.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niels Langeveld (Leiden University)
DTSTART:20200630T123000Z
DTEND:20200630T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/9
DESCRIPTION:Title: Continued fractions with two non integer digits\nby Niels Langevel
d (Leiden University) as part of One World Numeration seminar\n\n\nAbstrac
t\nIn this talk\, we will look at a family of continued fraction expansion
s for which the digits in the expansions can attain two different (typical
ly non-integer) values\, named $\\alpha_1$ and $\\alpha_2$ with $\\alpha_1
\\alpha_2 \\le 1/2$. If $\\alpha_1 \\alpha_2 < 1/2$ we can associate a dy
namical system to these expansions with a switch region and therefore with
lazy and greedy expansions. We will explore the parameter space and highl
ight certain values for which we can construct the natural extension (such
as a family for which the lowest digit cannot be followed by itself). We
end the talk with a list of open problems.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hajime Kaneko (University of Tsukuba)
DTSTART:20200707T123000Z
DTEND:20200707T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/10
DESCRIPTION:Title: Analogy of Lagrange spectrum related to geometric progressions\nb
y Hajime Kaneko (University of Tsukuba) as part of One World Numeration se
minar\n\n\nAbstract\nClassical Lagrange spectrum is defined by Diophantine
approximation properties of arithmetic progressions. The theory of Lagran
ge spectrum is related to number theory and symbolic dynamics. In our talk
we introduce significantly analogous results of Lagrange spectrum in unif
orm distribution theory of geometric progressions. In particular\, we disc
uss the geometric sequences whose common ratios are Pisot numbers. For stu
dying the fractional parts of geometric sequences\, we introduce certain n
umeration system. \n\nThis talk is based on a joint work with Shigeki Akiy
ama.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Attila Pethő (University of Debrecen)
DTSTART:20200714T123000Z
DTEND:20200714T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/11
DESCRIPTION:Title: On diophantine properties of generalized number systems - finite and
periodic representations\nby Attila Pethő (University of Debrecen) as
part of One World Numeration seminar\n\n\nAbstract\nIn this talk we inves
tigate elements with special patterns in their representations in number s
ystems in algebraic number fields. We concentrate on periodicity and on th
e representation of rational integers. We prove under natural assumptions
that there are only finitely many $S$-units whose representation is period
ic with a fixed period. We prove that the same holds for the set of values
of polynomials at rational integers.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derong Kong (Chongqing University)
DTSTART:20200623T123000Z
DTEND:20200623T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/12
DESCRIPTION:Title: Univoque bases of real numbers: local dimension\, Devil's staircase a
nd isolated points\nby Derong Kong (Chongqing University) as part of O
ne World Numeration seminar\n\n\nAbstract\nGiven a positive integer $M$ an
d a real number $x$\, let $U(x)$ be the set of all bases $q \\in (1\,M+1]$
such that $x$ has a unique $q$-expansion with respect to the alphabet $\\
{0\,1\,\\dots\,M\\}$. We will investigate the local dimension of $U(x)$ an
d prove a 'variation principle' for unique non-integer base expansions. We
will also determine the critical values and the topological structure of
$U(x)$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bill Mance (Adam Mickiewicz University in Poznań)
DTSTART:20200901T123000Z
DTEND:20200901T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/13
DESCRIPTION:Title: Hotspot Lemmas for Noncompact Spaces\nby Bill Mance (Adam Mickiew
icz University in Poznań) as part of One World Numeration seminar\n\n\nAb
stract\nWe will explore a correction of several previously claimed general
izations of the classical hotspot lemma. Specifically\, there is a common
mistake that has been repeated in proofs going back more than 50 years. Co
rrected versions of these theorems are increasingly important as there has
been more work in recent years focused on studying various generalization
s of the concept of a normal number to numeration systems with infinite di
git sets (for example\, various continued fraction expansions\, the Lürot
h series expansion and its generalizations\, and so on). Also\, highlighti
ng this (elementary) mistake may be helpful for those looking to study the
se numeration systems further and wishing to avoid some common pitfalls.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bing Li (South China University of Technology)
DTSTART:20200908T123000Z
DTEND:20200908T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/14
DESCRIPTION:Title: Some fractal problems in beta-expansions\nby Bing Li (South China
University of Technology) as part of One World Numeration seminar\n\n\nAb
stract\nFor greedy beta-expansions\, we study some fractal sets of real nu
mbers whose orbits under beta-transformation share some common properties.
For example\, the partial sum of the greedy beta-expansion converges with
the same order\, the orbit is not dense\, the orbit is always far from th
at of another point etc. The usual tool is to approximate the beta-transfo
rmation dynamical system by Markov subsystems. We also discuss the similar
problems for intermediate beta-expansions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Shallit (University of Waterloo)
DTSTART:20200915T123000Z
DTEND:20200915T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/15
DESCRIPTION:Title: Lazy Ostrowski Numeration and Sturmian Words\nby Jeffrey Shallit
(University of Waterloo) as part of One World Numeration seminar\n\n\nAbst
ract\nIn this talk I will discuss a new connection between the so-called "
lazy Ostrowski" numeration system\, and periods of the prefixes of Sturmia
n characteristic words. I will also give a relationship between periods an
d the so-called "initial critical exponent". This builds on work of Frid\,
Berthé-Holton-Zamboni\, Epifanio-Frougny-Gabriele-Mignosi\, and others\,
and is joint work with Narad Rampersad and Daniel Gabric.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yotam Smilansky (Rutgers University)
DTSTART:20200922T123000Z
DTEND:20200922T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/16
DESCRIPTION:Title: Multiscale Substitution Tilings\nby Yotam Smilansky (Rutgers Univ
ersity) as part of One World Numeration seminar\n\n\nAbstract\nMultiscale
substitution tilings are a new family of tilings of Euclidean space that a
re generated by multiscale substitution rules. Unlike the standard setup o
f substitution tilings\, which is a basic object of study within the aperi
odic order community and includes examples such as the Penrose and the pin
wheel tilings\, multiple distinct scaling constants are allowed\, and the
defining process of inflation and subdivision is a continuous one. Under a
certain irrationality assumption on the scaling constants\, this construc
tion gives rise to a new class of tilings\, tiling spaces and tiling dynam
ical system\, which are intrinsically different from those that arise in t
he standard setup. In the talk I will describe these new objects and discu
ss various structural\, geometrical\, statistical and dynamical results. B
ased on joint work with Yaar Solomon.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Maggioni (Leiden University)
DTSTART:20200929T123000Z
DTEND:20200929T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/17
DESCRIPTION:Title: Random matching for random interval maps\nby Marta Maggioni (Leid
en University) as part of One World Numeration seminar\n\n\nAbstract\nIn t
his talk we extend the notion of matching for deterministic transformation
s to random matching for random interval maps. For a large class of piecew
ise affine random systems of the interval\, we prove that this property of
random matching implies that any invariant density of a stationary measur
e is piecewise constant. We provide examples of random matching for a vari
ety of families of random dynamical systems\, that includes generalised be
ta-transformations\, continued fraction maps and a family of random maps p
roducing signed binary expansions. We finally apply the property of random
matching and its consequences to this family to study minimal weight expa
nsions. \nBased on a joint work with Karma Dajani and Charlene Kalle.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Veneziano (University of Genova)
DTSTART:20201006T123000Z
DTEND:20201006T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/18
DESCRIPTION:Title: Finiteness and periodicity of continued fractions over quadratic numb
er fields\nby Francesco Veneziano (University of Genova) as part of On
e World Numeration seminar\n\n\nAbstract\nWe consider continued fractions
with partial quotients in the ring of integers of a quadratic number field
$K$\; a particular example of these continued fractions is the $\\beta$-c
ontinued fraction introduced by Bernat. We show that for any quadratic Per
ron number $\\beta$\, the $\\beta$-continued fraction expansion of element
s in $\\mathbb{Q}(\\beta)$ is either finite of eventually periodic. We als
o show that for certain four quadratic Perron numbers $\\beta$\, the $\\be
ta$-continued fraction represents finitely all elements of the quadratic f
ield $\\mathbb{Q}(\\beta)$\, thus answering questions of Rosen and Bernat.
\nBased on a joint work with Zuzana Masáková and Tomáš Vávra.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kan Jiang (Ningbo University)
DTSTART:20201013T123000Z
DTEND:20201013T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/19
DESCRIPTION:Title: Representations of real numbers on fractal sets\nby Kan Jiang (Ni
ngbo University) as part of One World Numeration seminar\n\n\nAbstract\nTh
ere are many approaches which can represent real numbers. For instance\, t
he $\\beta$-expansions\, the continued fraction and so forth. Representati
ons of real numbers on fractal sets were pioneered by H. Steinhaus who pro
ved in 1917 that $C+C=[0\,2]$ and $C−C=[−1\,1]$\, where $C$ is the mid
dle-third Cantor set. Equivalently\, for any $x \\in [0\,2]$\, there exist
some $y\,z \\in C$ such that $x=y+z$. In this talk\, I will introduce sim
ilar results in terms of some fractal sets.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Surer (University of Natural Resources and Life Sciences\, Vi
enna)
DTSTART:20201020T123000Z
DTEND:20201020T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/20
DESCRIPTION:Title: Representations for complex numbers with integer digits\nby Paul
Surer (University of Natural Resources and Life Sciences\, Vienna) as part
of One World Numeration seminar\n\n\nAbstract\nIn this talk we present th
e zeta-expansion as a complex version of the well-known beta-expansion. It
allows us to expand complex numbers with respect to a complex base by usi
ng integer digits. Our concepts fits into the framework of the recently pu
blished rotational beta-expansions. But we also establish relations with p
iecewise affine maps of the torus and with shift radix systems.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mélodie Andrieu (Aix-Marseille University)
DTSTART:20201027T133000Z
DTEND:20201027T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/21
DESCRIPTION:Title: A Rauzy fractal unbounded in all directions of the plane\nby Mél
odie Andrieu (Aix-Marseille University) as part of One World Numeration se
minar\n\n\nAbstract\nUntil 2001 it was believed that\, as for Sturmian wor
ds\, the imbalance of Arnoux-Rauzy words was bounded - or at least finite.
Cassaigne\, Ferenczi and Zamboni disproved this conjecture by constructin
g an Arnoux-Rauzy word with infinite imbalance\, i.e. a word whose broken
line deviates regularly and further and further from its average direction
. Today\, we hardly know anything about the geometrical and topological pr
operties of these unbalanced Rauzy fractals. The Oseledets theorem suggest
s that these fractals are contained in a strip of the plane: indeed\, if t
he Lyapunov exponents of the matricial product associated with the word ex
ist\, one of these exponents at least is nonpositive since their sum equal
s zero. This talk aims at disproving this belief.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomáš Vávra (University of Waterloo)
DTSTART:20201103T133000Z
DTEND:20201103T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/22
DESCRIPTION:Title: Distinct unit generated number fields and finiteness in number system
s\nby Tomáš Vávra (University of Waterloo) as part of One World Num
eration seminar\n\n\nAbstract\nA distinct unit generated field is a number
field K such that every algebraic integer of the field is a sum of distin
ct units. In 2015\, Dombek\, Masáková\, and Ziegler studied totally comp
lex quartic fields\, leaving 8 cases unresolved. Because in this case ther
e is only one fundamental unit $u$\, their method involved the study of fi
niteness in positional number systems with base u and digits arising from
the roots of unity in $K$.\n \nFirst\, we consider a more general problem
of positional representations with base beta with an arbitrary digit alpha
bet $D$. We will show that it is decidable whether a given pair $(\\beta\,
D)$ allows eventually periodic or finite representations of elements of $
O_K$.\n \nWe are then able to prove the conjecture that the 8 remaining ca
ses indeed are distinct unit generated.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pieter Allaart (University of North Texas)
DTSTART:20201110T133000Z
DTEND:20201110T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/23
DESCRIPTION:Title: On the smallest base in which a number has a unique expansion\nby
Pieter Allaart (University of North Texas) as part of One World Numeratio
n seminar\n\n\nAbstract\nFor $x>0$\, let $U(x)$ denote the set of bases $q
\\in (1\,2]$ such that $x$ has a unique expansion in base $q$ over the al
phabet $\\{0\,1\\}$\, and let $f(x)=\\inf U(x)$. I will explain that the f
unction $f(x)$ has a very complicated structure: it is highly discontinuou
s and has infinitely many infinite level sets. I will describe an algorith
m for numerically computing $f(x)$ that often gives the exact value in jus
t a small finite number of steps. The Komornik-Loreti constant\, which is
$f(1)$\, will play a central role in this talk. This is joint work with De
rong Kong\, and builds on previous work by Kong (Acta Math. Hungar. 150(1)
:194--208\, 2016).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacques Sakarovitch (Irif\, CNRS\, and Télécom Paris)
DTSTART:20201117T133000Z
DTEND:20201117T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/24
DESCRIPTION:Title: The carry propagation of the successor function\nby Jacques Sakar
ovitch (Irif\, CNRS\, and Télécom Paris) as part of One World Numeration
seminar\n\n\nAbstract\nGiven any numeration system\, the carry propagatio
n at an integer $N$ is the number of digits that change between the repres
entation of $N$ and $N+1$. The carry propagation of the numeration system
as a whole is the average carry propagations at the first $N$ integers\, a
s $N$ tends to infinity\, if this limit exists. \n\nIn the case of the usu
al base $p$ numeration system\, it can be shown that the limit indeed exis
ts and is equal to $p/(p-1)$. We recover a similar value for those numerat
ion systems we consider and for which the limit exists. \n\nThe problem is
less the computation of the carry propagation than the proof of its exist
ence. We address it for various kinds of numeration systems: abstract nume
ration systems\, rational base numeration systems\, greedy numeration syst
ems and beta-numeration. This problem is tackled with three different type
s of techniques: combinatorial\, algebraic\, and ergodic\, each of them be
ing relevant for different kinds of numeration systems. \n\nThis work has
been published in Advances in Applied Mathematics 120 (2020). In this talk
\, we shall focus on the algebraic and ergodic methods. \n\nJoint work wit
h V. Berthé (Irif)\, Ch. Frougny (Irif)\, and M. Rigo (Univ. Liège).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Barnsley (Australian National University)
DTSTART:20201201T133000Z
DTEND:20201201T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/25
DESCRIPTION:Title: Rigid fractal tilings\nby Michael Barnsley (Australian National U
niversity) as part of One World Numeration seminar\n\n\nAbstract\nI will d
escribe recent work\, joint with Louisa Barnsley and Andrew Vince\, concer
ning a symbolic approach to self-similar tilings. This approach uses graph
-directed iterated function systems to analyze both classical tilings and
also generalized tilings of what may be unbounded fractal subsets of $\\ma
thbb{R}^n$. A notion of rigid tiling systems is defined. Our key theorem s
tates that when the system is rigid\, all the conjugacies of the tilings c
an be described explicitly. In the seminar I hope to prove this for the ca
se of standard IFSs.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa)
DTSTART:20201208T133000Z
DTEND:20201208T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/26
DESCRIPTION:Title: Limit theorems on counting large continued fraction digits\nby Ta
nja Isabelle Schindler (Scuola Normale Superiore di Pisa) as part of One W
orld Numeration seminar\n\n\nAbstract\nWe establish a central limit theore
m for counting large continued fraction digits $(a_n)$\, that is\, we coun
t occurrences $\\{a_n>b_n\\}$\, where $(b_n)$ is a sequence of positive in
tegers. Our result improves a similar result by Philipp\, which additional
ly assumes that bn tends to infinity. Moreover\, we also show this kind of
central limit theorem for counting the number of occurrences entries such
that the continued fraction entry lies between $d_n$ and $d_n(1+1/c_n)$ f
or given sequences $(c_n)$ and $(d_n)$. For such intervals we also give a
refinement of the famous Borel–Bernstein theorem regarding the event tha
t the nth continued fraction digit lying infinitely often in this interval
. As a side result\, we explicitly determine the first $\\phi$-mixing coef
ficient for the Gauss system - a result we actually need to improve Philip
p's theorem. This is joint work with Marc Kesseböhmer.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lukas Spiegelhofer (Montanuniversität Leoben)
DTSTART:20201215T133000Z
DTEND:20201215T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/27
DESCRIPTION:Title: The digits of $n+t$\nby Lukas Spiegelhofer (Montanuniversität Le
oben) as part of One World Numeration seminar\n\n\nAbstract\nWe study the
binary sum-of-digits function $s_2$ under addition of a constant $t$.\nFor
each integer $k$\, we are interested in the asymptotic density $\\delta(k
\,t)$ of integers $t$ such that $s_2(n+t)-s_2(n)=k$.\nIn this talk\, we co
nsider the following two questions. \n\n(1) Do we have \\[ c_t=\\delta(0
\,t)+\\delta(1\,t)+\\cdots>1/2? \\]\nThis is a conjecture due to T. W. Cu
sick (2011). \n\n(2) What does the probability distribution defined by $k\
\mapsto \\delta(k\,t)$ look like?\n\nWe prove that indeed $c_t>1/2$ if the
binary expansion of $t$ contains at least $M$ blocks of contiguous ones\,
where $M$ is effective.\nOur second theorem states that $\\delta(j\,t)$ u
sually behaves like a normal distribution\, which extends a result by Emme
and Hubert (2018).\n\nThis is joint work with Michael Wallner (TU Wien).\
n
LOCATION:https://stable.researchseminars.org/talk/OWNS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claire Merriman (Ohio State University)
DTSTART:20210105T133000Z
DTEND:20210105T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/28
DESCRIPTION:Title: $\\alpha$-odd continued fractions\nby Claire Merriman (Ohio State
University) as part of One World Numeration seminar\n\n\nAbstract\nThe st
andard continued fraction algorithm come from the Euclidean algorithm. We
can also describe this algorithm using a dynamical system of $[0\,1)$\, wh
ere the transformation that takes $x$ to the fractional part of $1/x$ is s
aid to generate the continued fraction expansion of $x$. From there\, we a
sk two questions: What happens to the continued fraction expansion when we
change the domain to something other than $[0\,1)$? What happens to the d
ynamical system when we impose restrictions on the continued fraction expa
nsion\, such as finding the nearest odd integer instead of the floor? This
talk will focus on the case where we first restrict to odd integers\, the
n start shifting the domain $[\\alpha-2\, \\alpha)$.\n \nThis talk is base
d on joint work with Florin Boca and animations done by Xavier Ding\, Gust
av Jennetten\, and Joel Rozhon as part of an Illinois Geometry Lab project
.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Kempton (University of Manchester)
DTSTART:20210119T133000Z
DTEND:20210119T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/29
DESCRIPTION:Title: Bernoulli Convolutions and Measures on the Spectra of Algebraic Integ
ers\nby Tom Kempton (University of Manchester) as part of One World Nu
meration seminar\n\n\nAbstract\nGiven an algebraic integer $\\beta$ and al
phabet $A=\\{-1\,0\,1\\}$\, the spectrum of $\\beta$ is the set \n$$\\Sigm
a(\\beta) :=\\bigg\\{\\sum_{i=1}^n a_i\\beta^i : n\\in\\mathbb N\, a_i\\in
A\\bigg\\}.$$\nIn the case that $\\beta$ is Pisot one can study the spect
rum of $\\beta$ dynamically using substitutions or cut and project schemes
\, and this allows one to see lots of local structure in the spectrum. The
re are higher dimensional analogues for other algebraic integers.\n\nIn th
is talk we will define a random walk on the spectrum of $\\beta$ and show
how\, with appropriate renormalisation\, this leads to an infinite station
ary measure on the spectrum. This measure has local structure analagous to
that of the spectrum itself. Furthermore\, this measure has deep links wi
th the Bernoulli convolution\, and in particular new criteria for the abso
lute continuity of Bernoulli convolutions can be stated in terms of the er
godic properties of these measures.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Carminati (Università di Pisa)
DTSTART:20210126T133000Z
DTEND:20210126T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/30
DESCRIPTION:Title: Prevalence of matching for families of continued fraction algorithms:
old and new results\nby Carlo Carminati (Università di Pisa) as part
of One World Numeration seminar\n\n\nAbstract\nWe will give an overview o
f the phenomenon of matching\, which was first observed in the family of N
akada's $\\alpha$-continued fractions\, but is also encountered in other f
amilies of continued fraction algorithms.\n\nOur main focus will be the ma
tching property for the family of Ito-Tanaka continued fractions: we will
discuss the analogies with Nakada's case\n(such as prevalence of matching)
\, but also some unexpected features which are peculiar of this case.\n\nT
he core of the talk is about some recent results obtained in collaboration
with Niels Langeveld and Wolfgang Steiner.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Petite (Université de Picardie Jules Verne)
DTSTART:20210202T133000Z
DTEND:20210202T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/31
DESCRIPTION:Title: Interplay between finite topological rank minimal Cantor systems\, $S
$-adic subshifts and their complexity\nby Samuel Petite (Université d
e Picardie Jules Verne) as part of One World Numeration seminar\n\n\nAbstr
act\nThe family of minimal Cantor systems of finite topological rank inclu
des Sturmian subshifts\, coding of interval exchange transformations\, odo
meters and substitutive subshifts. They are known to have dynamical rigidi
ty properties. In a joint work with F. Durand\, S. Donoso and A. Maass\, w
e provide a combinatorial characterization of such subshifts in terms of S
-adic systems. This enables to obtain some links with the factor complexit
y function and some new rigidity properties depending on the rank of the s
ystem.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clemens Müllner (TU Wien)
DTSTART:20210209T133000Z
DTEND:20210209T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/32
DESCRIPTION:Title: Multiplicative automatic sequences\nby Clemens Müllner (TU Wien)
as part of One World Numeration seminar\n\n\nAbstract\nIt was shown by Ma
riusz Lemańczyk and the author that automatic sequences are orthogonal to
bounded and aperiodic multiplicative functions. This is a manifestation o
f the disjointedness of additive and multiplicative structures. We continu
e this path by presenting in this talk a complete classification of comple
x-valued sequences which are both multiplicative and automatic. This shows
that the intersection of these two worlds has a very special (and simple)
form. This is joint work with Mariusz Lemańczyk and Jakub Konieczny.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gerardo González Robert (Universidad Nacional Autónoma de Méxic
o)
DTSTART:20210216T133000Z
DTEND:20210216T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/33
DESCRIPTION:Title: Good's Theorem for Hurwitz Continued Fractions\nby Gerardo Gonzá
lez Robert (Universidad Nacional Autónoma de México) as part of One Worl
d Numeration seminar\n\n\nAbstract\nIn 1887\, Adolf Hurwitz introduced a s
imple procedure to write any complex number as a continued fraction with G
aussian integers as partial denominators and with partial numerators equal
to 1. While similarities between regular and Hurwitz continued fractions
abound\, there are important differences too (for example\, as shown in 19
74 by R. Lakein\, Serret's theorem on equivalent numbers does not hold in
the complex case). In this talk\, after giving a short overview of the the
ory of Hurwitz continued fractions\, we will state and sketch the proof of
a complex version of I. J. Good's theorem on the Hausdorff dimension of t
he set of real numbers whose regular continued fraction tends to infinity.
Finally\, we will discuss some open problems.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seulbee Lee (Scuola Normale Superiore di Pisa)
DTSTART:20210223T133000Z
DTEND:20210223T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/34
DESCRIPTION:Title: Odd-odd continued fraction algorithm\nby Seulbee Lee (Scuola Norm
ale Superiore di Pisa) as part of One World Numeration seminar\n\n\nAbstra
ct\nThe classical continued fraction gives the best approximating rational
numbers of an irrational number. We define a new continued fraction\, say
odd-odd continued fraction\, which gives the best approximating rational
numbers whose numerators and denominators are odd. We see that a jump tran
sformation associated to the Romik map induces the odd-odd continued fract
ion. We discuss properties of the odd-odd continued fraction expansions. T
his is joint work with Dong Han Kim and Lingmin Liao.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitaly Bergelson (Ohio State University)
DTSTART:20210302T150000Z
DTEND:20210302T160000Z
DTSTAMP:20260404T131146Z
UID:OWNS/35
DESCRIPTION:Title: Normal sets in $(\\mathbb{ℕ}\,+)$ and $(\\mathbb{N}\,\\times)$\
nby Vitaly Bergelson (Ohio State University) as part of One World Numerati
on seminar\n\n\nAbstract\nWe will start with discussing the general idea o
f a normal set in a countable cancellative amenable semigroup\, which was
introduced and developed in the recent paper "A fresh look at the notion o
f normality" (joint work with Tomas Downarowicz and Michał Misiurewicz).
We will move then to discussing and juxtaposing combinatorial and Diophant
ine properties of normal sets in semigroups $(\\mathbb{ℕ}\,+)$ and $(\\m
athbb{N}\,\\times)$. We will conclude the lecture with a brief review of s
ome interesting open problems.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Priebe Frank (Vassar College)
DTSTART:20210309T133000Z
DTEND:20210309T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/36
DESCRIPTION:Title: The flow view and infinite interval exchange transformation of a reco
gnizable substitution\nby Natalie Priebe Frank (Vassar College) as par
t of One World Numeration seminar\n\n\nAbstract\nA flow view is the graph
of a measurable conjugacy between a substitution or S-adic subshift or til
ing space and an exchange of infinitely many intervals in [0\,1]. The natu
ral refining sequence of partitions of the sequence space is transferred t
o [0\,1] with Lebesgue measure using a canonical addressing scheme\, a fix
ed dual substitution\, and a shift-invariant probability measure. On the f
low view\, sequences are shown horizontally at a height given by their ima
ge under conjugacy.\n\nIn this talk I'll explain how it all works and stat
e some results and questions. There will be pictures.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Skripchenko (Higher School of Economics)
DTSTART:20210316T133000Z
DTEND:20210316T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/37
DESCRIPTION:Title: Double rotations and their ergodic properties\nby Alexandra Skrip
chenko (Higher School of Economics) as part of One World Numeration semina
r\n\n\nAbstract\nDouble rotations are the simplest subclass of interval tr
anslation mappings. A double rotation is of finite type if its attractor i
s an interval and of infinite type if it is a Cantor set. It is easy to se
e that the restriction of a double rotation of finite type to its attracto
r is simply a rotation. It is known due to Suzuki - Ito - Aihara and Bruin
- Clark that double rotations of infinite type are defined by a subset of
zero measure in the parameter set. We introduce a new renormalization pro
cedure on double rotations\, which is reminiscent of the classical Rauzy i
nduction. Using this renormalization we prove that the set of parameters w
hich induce infinite type double rotations has Hausdorff dimension strictl
y smaller than 3. Moreover\, we construct a natural invariant measure supp
orted on these parameters and show that\, with respect to this measure\, a
lmost all double rotations are uniquely ergodic. In my talk I plan to outl
ine this proof that is based on the recent result by Ch. Fougeron for simp
licial systems. I also hope to discuss briefly some challenging open quest
ions and further research plans related to double rotations. \n\nThe talk
is based on a joint work with Mauro Artigiani\, Charles Fougeron and Pasca
l Hubert.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Godofredo Iommi (Pontificia Universidad Católica de Chile)
DTSTART:20210323T133000Z
DTEND:20210323T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/38
DESCRIPTION:Title: Arithmetic averages and normality in continued fractions\nby Godo
fredo Iommi (Pontificia Universidad Católica de Chile) as part of One Wor
ld Numeration seminar\n\n\nAbstract\nEvery real number can be written as a
continued fraction. There exists a dynamical system\, the Gauss map\, tha
t acts as the shift in the expansion. In this talk\, I will comment on the
Hausdorff dimension of two types of sets: one of them defined in terms of
arithmetic averages of the digits in the expansion and the other related
to (continued fraction) normal numbers. In both cases\, the non compactnes
s that steams from the fact that we use countable many partial quotients i
n the continued fraction plays a fundamental role. Some of the results are
joint work with Thomas Jordan and others together with Aníbal Velozo.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Drmota (TU Wien)
DTSTART:20210330T123000Z
DTEND:20210330T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/39
DESCRIPTION:Title: (Logarithmic) Densities for Automatic Sequences along Primes and Squa
res\nby Michael Drmota (TU Wien) as part of One World Numeration semin
ar\n\n\nAbstract\nIt is well known that the every letter $\\alpha$ of an a
utomatic sequence $a(n)$ has\na logarithmic density -- and it can be decid
ed when this logarithmic density is actually a density.\nFor example\, the
letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequenc
es $1/2$.\n[The Thue-Morse sequence is the binary sum-of-digits functions
modulo 2.]\n\nThe purpose of this talk is to present a corresponding resul
t for subsequences of general\nautomatic sequences along primes and square
s. This is a far reaching generalization of two breakthrough\nresults of M
auduit and Rivat from 2009 and 2010\, where they solved two conjectures by
Gelfond\non the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where
$p_n$ denotes the sequence of primes).\n\nMore technically\, one has to de
velop a method to transfer density results for primitive automatic\nsequen
ces to logarithmic-density results for general automatic sequences. Then a
s an application\none can deduce that the logarithmic densities of any aut
omatic sequence along squares\n$(n^2)_{n\\geq 0}$ and primes $(p_n)_{n\\ge
q 1}$ exist and are computable.\nFurthermore\, if densities exist then the
y are (usually) rational.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Mitchell (University of Birmingham)
DTSTART:20210413T123000Z
DTEND:20210413T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/40
DESCRIPTION:Title: Measure theoretic entropy of random substitutions\nby Andrew Mitc
hell (University of Birmingham) as part of One World Numeration seminar\n\
n\nAbstract\nRandom substitutions and their associated subshifts provide a
model for structures that exhibit both long range order and positive topo
logical entropy. In this talk we discuss the entropy of a large class of e
rgodic measures\, known as frequency measures\, that arise naturally from
random substitutions. We introduce a new measure of complexity\, namely me
asure theoretic inflation word entropy\, and discuss its relationship to m
easure theoretic entropy. This new measure of complexity provides a framew
ork for the systematic study of measure theoretic entropy for random subst
itution subshifts. \n\nAs an application of our results\, we obtain closed
form formulas for the entropy of frequency measures for a wide range of r
andom substitution subshifts and show that in many cases there exists a fr
equency measure of maximal entropy. Further\, for a class of random substi
tution subshifts\, we show that this measure is the unique measure of maxi
mal entropy.\n\nThis talk is based on joint work with P. Gohlke\, D. Rust\
, and T. Samuel.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayreena Bakhtawar (La Trobe University)
DTSTART:20210420T123000Z
DTEND:20210420T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/41
DESCRIPTION:Title: Metrical theory for the set of points associated with the generalized
Jarnik-Besicovitch set\nby Ayreena Bakhtawar (La Trobe University) as
part of One World Numeration seminar\n\n\nAbstract\nFrom Lagrange's (1770
) and Legendre's (1808) results we conclude that to find good rational app
roximations to an irrational number we only need to focus on its convergen
ts. Let $[a_1(x)\,a_2(x)\,\\dots]$ be the continued fraction expansion of
a real number $x \\in [0\,1)$. The Jarnik-Besicovitch set in terms of cont
inued fraction consists of all those $x \\in [0\,1)$ which satisfy $a_{n+1
}(x) \\ge e^{\\tau\\\, (\\log|T'x|+⋯+\\log|T'(T^{n-1}x)|)}$ for infinite
ly many $n \\in \\mathbb{N}$\, where $a_{n+1}(x)$ is the $(n+1)$-th partia
l quotient of $x$ and $T$ is the Gauss map. In this talk\, I will focus on
determining the Hausdorff dimension of the set of real numbers $x \\in [0
\,1)$ such that for any $m \\in \\mathbb{N}$ the following holds for infin
itely many $n \\in \\mathbb{N}$: $a_{n+1}(x) a_{n+2}(x) \\cdots a_{n+m}(x)
\\ge e^{τ(x)\\\, (f(x)+⋯+f(T^{n-1}x))}$\, where $f$ and $\\tau$ are po
sitive continuous functions. Also we will see that for appropriate choices
of $m$\, $\\tau(x)$ and $f(x)$ our result implies various classical resul
ts including the famous Jarnik-Besicovitch theorem.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (CNRS\, Université Claude Bernard Lyon 1)
DTSTART:20210427T123000Z
DTEND:20210427T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/42
DESCRIPTION:Title: Expansions of numbers in multiplicatively independent bases: Furstenb
erg's conjecture and finite automata\nby Boris Adamczewski (CNRS\, Uni
versité Claude Bernard Lyon 1) as part of One World Numeration seminar\n\
n\nAbstract\nIt is commonly expected that expansions of numbers in multipl
icatively independent bases\, such as 2 and 10\, should have no common str
ucture. However\, it seems extraordinarily difficult to confirm this naive
heuristic principle in some way or another. In the late 1960s\, Furstenbe
rg suggested a series of conjectures\, which became famous and aim to capt
ure this heuristic. The work I will discuss in this talk is motivated by o
ne of these conjectures. Despite recent remarkable progress by Shmerkin an
d Wu\, it remains totally out of reach of the current methods. While Furst
enberg’s conjectures take place in a dynamical setting\, I will use inst
ead the language of automata theory to formulate some related problems tha
t formalize and express in a different way the same general heuristic. I w
ill explain how the latter can be solved thanks to some recent advances in
Mahler’s method\; a method in transcendental number theory initiated by
Mahler at the end of the 1920s. This a joint work with Colin Faverjon.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tushar Das (University of Wisconsin - La Crosse)
DTSTART:20210504T140000Z
DTEND:20210504T150000Z
DTSTAMP:20260404T131146Z
UID:OWNS/43
DESCRIPTION:Title: Hausdorff Hensley Good & Gauss\nby Tushar Das (University of Wisc
onsin - La Crosse) as part of One World Numeration seminar\n\n\nAbstract\n
Several participants of the One World Numeration Seminar (OWNS) will know
Hensley's haunting bounds (c. 1990) for the dimension of irrationals whose
regular continued fraction expansion partial quotients are all at most N\
; while some might remember Good's great bounds (c. 1940) for the dimensio
n of irrationals whose partial quotients are all at least N. We will repor
t on relatively recent results in https://arxiv.org/abs/2007.10554 that al
low one to extend such fabulous formulae to unexpected expansions. Our tec
hnology may be utilized to study various systems arising from numeration\,
dynamics\, or geometry. The talk will be accessible to students and beyon
d\, and I hope to present a sampling of open questions and research direct
ions that await exploration.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giulio Tiozzo (University of Toronto)
DTSTART:20210511T123000Z
DTEND:20210511T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/44
DESCRIPTION:Title: The bifurcation locus for numbers of bounded type\nby Giulio Tioz
zo (University of Toronto) as part of One World Numeration seminar\n\n\nAb
stract\nWe define a family $B(t)$ of compact subsets of the unit interval
which provides a filtration of the set of numbers whose continued fraction
expansion has bounded digits. This generalizes to a continuous family the
well-known sets of numbers whose continued fraction expansion is bounded
above by a fixed integer. \n\nWe study how the set $B(t)$ changes as the p
arameter $t$ ranges in $[0\,1]$\, and describe precisely the bifurcations
that occur as the parameters change. Further\, we discuss continuity prope
rties of the Hausdorff dimension of $B(t)$ and its regularity. \n\nFinally
\, we establish a precise correspondence between these bifurcations and \n
the bifurcations for the classical family of real quadratic polynomials. \
n\nJoint with C. Carminati.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Vandehey (University of Texas at Tyler)
DTSTART:20210518T123000Z
DTEND:20210518T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/45
DESCRIPTION:Title: Solved and unsolved problems in normal numbers\nby Joseph Vandehe
y (University of Texas at Tyler) as part of One World Numeration seminar\n
\n\nAbstract\nWe will survey a variety of problems on normal numbers\, som
e old\, some new\, some solved\, and some unsolved\, in the hope of spurri
ng some new directions of study. Topics will include constructions of norm
al numbers\, normality in two different systems simultaneously\, normality
seen through the lens of informational or logical complexity\, and more.\
n
LOCATION:https://stable.researchseminars.org/talk/OWNS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charles Fougeron (Université de Paris)
DTSTART:20210525T123000Z
DTEND:20210525T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/46
DESCRIPTION:Title: Dynamics of simplicial systems and multidimensional continued fractio
n algorithms\nby Charles Fougeron (Université de Paris) as part of On
e World Numeration seminar\n\n\nAbstract\nMotivated by the richness of the
Gauss algorithm which allows to efficiently compute the best approximatio
ns of a real number by rationals\, many mathematicians have suggested gene
ralisations to study Diophantine approximations of vectors in higher dimen
sions. Examples include Poincaré's algorithm introduced at the end of the
19th century or those of Brun and Selmer in the middle of the 20th centur
y. Since the beginning of the 90's to the present day\, there has been man
y works studying the convergence and dynamics of these multidimensional co
ntinued fraction algorithms. In particular\, Schweiger and Broise have sho
wn that the approximation sequence built using Selmer and Brun algorithms
converge to the right vector with an extra ergodic property. On the other
hand\, Nogueira demonstrated that the algorithm proposed by Poincaré almo
st never converges. \n\nStarting from the classical case of Farey's algori
thm\, which is an "additive" version of Gauss's algorithm\, I will present
a combinatorial point of view on these algorithms which allows to us to u
se a random walk approach. In this model\, taking a random vector for the
Lebesgue measure will correspond to following a random walk with memory in
a labelled graph called symplicial system. The laws of probability for th
is random walk are elementary and we can thus develop probabilistic techni
ques to study their generic dynamical behaviour. This will lead us to desc
ribe a purely graph theoretic criterion to check the convergence of a cont
inued fraction algorithm.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bastián Espinoza (Université de Picardie Jules Verne and Univers
idad de Chile)
DTSTART:20210601T123000Z
DTEND:20210601T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/47
DESCRIPTION:Title: Automorphisms and factors of finite topological rank systems\nby
Bastián Espinoza (Université de Picardie Jules Verne and Universidad de
Chile) as part of One World Numeration seminar\n\n\nAbstract\nFinite topol
ogical rank systems are a type of minimal S-adic subshift that includes ma
ny of the classical minimal systems of zero entropy (e.g. linearly recurre
nt subshifts\, interval exchanges and some Toeplitz sequences). In this ta
lk I am going to present results concerning the number of automorphisms an
d factors of systems of finite topological rank\, as well as closure prope
rties of this class with respect to factors and related combinatorial oper
ations.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shigeki Akiyama (University of Tsukuba)
DTSTART:20210608T123000Z
DTEND:20210608T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/48
DESCRIPTION:Title: Counting balanced words and related problems\nby Shigeki Akiyama
(University of Tsukuba) as part of One World Numeration seminar\n\n\nAbstr
act\nBalanced words and Sturmian words are ubiquitous and appear in the in
tersection of many areas of mathematics. In this talk\, I try to explain a
n idea of S. Yasutomi to study finite balanced words. His method gives a n
ice way to enumerate number of balanced words of given length\, slope and
intercept. Applying this idea\, we can obtain precise asymptotic formula f
or balanced words. The result is connected to some classical topics in num
ber theory\, such as Farey fraction\, Riemann Hypothesis and Large sieve i
nequality.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART:20210615T123000Z
DTEND:20210615T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/49
DESCRIPTION:Title: Dyadic approximation in the Cantor set\nby Sam Chow (University o
f Warwick) as part of One World Numeration seminar\n\n\nAbstract\nWe inves
tigate the approximation rate of a typical element of the Cantor set by dy
adic rationals. This is a manifestation of the times-two-times-three pheno
menon\, and is joint work with Demi Allen and Han Yu.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lingmin Liao (Université Paris-Est Créteil Val de Marne)
DTSTART:20210622T123000Z
DTEND:20210622T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/50
DESCRIPTION:Title: Simultaneous Diophantine approximation of the orbits of the dynamical
systems x2 and x3\nby Lingmin Liao (Université Paris-Est Créteil Va
l de Marne) as part of One World Numeration seminar\n\n\nAbstract\nWe stud
y the sets of points whose orbits of the dynamical systems x2 and x3 simul
taneously approach to a given point\, with a given speed. A zero-one law f
or the Lebesgue measure of such sets is established. The Hausdorff dimensi
ons are also determined for some special speeds. One dimensional formula a
mong them is established under the abc conjecture. At the same time\, we a
lso study the Diophantine approximation of the orbits of a diagonal matrix
transformation of a torus\, for which the properties of the (negative) be
ta transformations are involved. This is a joint work with Bing Li\, Sanju
Velani and Evgeniy Zorin.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Polina Vytnova (University of Warwick)
DTSTART:20210629T123000Z
DTEND:20210629T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/51
DESCRIPTION:Title: Hausdorff dimension of Gauss-Cantor sets and their applications to th
e study of classical Markov spectrum\nby Polina Vytnova (University of
Warwick) as part of One World Numeration seminar\n\n\nAbstract\nThe class
ical Lagrange and Markov spectra are subsets of the real line which arise
in connection with some problems in theory Diophantine approximation theor
y. In 1921 O. Perron gave a definition in terms of continued fractions\, w
hich allowed to study the Markov and Lagrange spectra using limit sets of
iterated function schemes. \n\nIn this talk we will see how the first tran
sition point\, where the Markov spectra acquires the full measure can be c
omputed by the means of estimating Hausdorff dimension of the certain Gaus
s-Cantor sets. \n\nThe talk is based on a joint work with C. Matheus\, C.
G. Moreira and M. Pollicott.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niclas Technau (University of Wisconsin - Madison)
DTSTART:20210706T123000Z
DTEND:20210706T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/52
DESCRIPTION:Title: Littlewood and Duffin-Schaeffer-type problems in diophantine approxim
ation\nby Niclas Technau (University of Wisconsin - Madison) as part o
f One World Numeration seminar\n\n\nAbstract\nGallagher's theorem describe
s the multiplicative diophantine approximation rate of a typical vector. R
ecently Sam Chow and I establish a fully-inhomogeneous version of Gallaghe
r's theorem\, and a diophantine fibre refinement. In this talk I outline t
he proof\, and the tools involved in it.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Karpenkov (University of Liverpool)
DTSTART:20210907T123000Z
DTEND:20210907T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/53
DESCRIPTION:Title: On Hermite's problem\, Jacobi-Perron type algorithms\, and Dirichlet
groups\nby Oleg Karpenkov (University of Liverpool) as part of One Wor
ld Numeration seminar\n\n\nAbstract\nIn this talk we introduce a new modif
ication of the Jacobi-Perron algorithm in the three dimensional case. This
algorithm is periodic for the case of totally-real conjugate cubic vector
s. To the best of our knowledge this is the first Jacobi-Perron type algor
ithm for which the cubic periodicity is proven. This provides an answer in
the totally-real case to the question of algebraic periodicity for cubic
irrationalities posed in 1848 by Ch.Hermite.\n\nWe will briefly discuss a
new approach which is based on geometry of numbers. In addition we point o
ut one important application of Jacobi-Perron type algorithms to the compu
tation of independent elements in the maximal groups of commuting matrices
of algebraic irrationalities.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henna Koivusalo (University of Vienna)
DTSTART:20200602T123000Z
DTEND:20200602T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/54
DESCRIPTION:Title: Linear repetition in polytopal cut and project sets\nby Henna Koi
vusalo (University of Vienna) as part of One World Numeration seminar\n\n\
nAbstract\nCut and project sets are aperiodic point patterns obtained by p
rojecting an irrational slice of the integer lattice to a subspace. One wa
y of classifying aperiodic sets is to study repetition of finite patterns\
, where sets with linear pattern repetition can be considered as the most
ordered aperiodic sets. \nRepetitivity of a cut and project set depends on
the slope and shape of the irrational slice. The cross-section of the sli
ce is known as the window. In an earlier work it was shown that for cut an
d project sets with a cube window\, linear repetitivity holds if and only
if the following two conditions are satisfied: (i) the set has minimal com
plexity and (ii) the irrational slope satisfies a certain Diophantine cond
ition. In a new joint work with Jamie Walton\, we give a generalisation of
this result for other polytopal windows\, under mild geometric conditions
. A key step in the proof is a decomposition of the cut and project scheme
\, which allows us to make sense of condition (ii) for general polytopal w
indows.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Jackson (University of North Texas)
DTSTART:20210914T123000Z
DTEND:20210914T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/55
DESCRIPTION:Title: Descriptive complexity in numeration systems\nby Steve Jackson (U
niversity of North Texas) as part of One World Numeration seminar\n\n\nAbs
tract\nDescriptive set theory gives a means of calibrating the complexity
of sets\, and we focus on some sets occurring in numerations systems. Also
\, the descriptive complexity of the difference of two sets gives a notion
of the logical independence of the sets. A classic result of Ki and Linto
n says that the set of normal numbers for a given base is a $\\boldsymbol{
\\Pi}^0_3$ complete set. In work with Airey\, Kwietniak\, and Mance we ext
end to other numerations systems such as continued fractions\, $\\beta$-ex
pansions\, and GLS expansions. In work with Mance and Vandehey we show tha
t the numbers which are continued fraction normal but not base $b$ normal
is complete at the expected level of $D_2(\\boldsymbol{\\Pi}^0_3)$. An imm
ediate corollary is that this set is uncountable\, a result (due to Vandeh
ey) only known previously assuming the generalized Riemann hypothesis.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Siskaki (University of Illinois at Urbana-Champaign)
DTSTART:20210921T123000Z
DTEND:20210921T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/56
DESCRIPTION:Title: The distribution of reduced quadratic irrationals arising from contin
ued fraction expansions\nby Maria Siskaki (University of Illinois at U
rbana-Champaign) as part of One World Numeration seminar\n\n\nAbstract\nIt
is known that the reduced quadratic irrationals arising from regular cont
inued fraction expansions are uniformly distributed when ordered by their
length with respect to the Gauss measure. In this talk\, I will describe a
number theoretical approach developed by Kallies\, Ozluk\, Peter and Snyd
er\, and then by Boca\, that gives the error in the asymptotic behavior of
this distribution. Moreover\, I will present the respective result for th
e distribution of reduced quadratic irrationals that arise from even (join
t work with F. Boca) and odd continued fractions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Hieronymi (Universität Bonn)
DTSTART:20210928T123000Z
DTEND:20210928T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/57
DESCRIPTION:Title: A strong version of Cobham's theorem\nby Philipp Hieronymi (Unive
rsität Bonn) as part of One World Numeration seminar\n\n\nAbstract\nLet $
k\,l>1$ be two multiplicatively independent integers. A subset $X$ of $\\m
athbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $
X$ is recognized by some finite automaton. Cobham’s famous theorem state
s that a subset of the natural numbers is both $k$-recognizable and $l$-re
cognizable if and only if it is Presburger-definable (or equivalently: sem
ilinear). We show the following strengthening. Let $X$ be $k$-recognizable
\, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburg
er-definable. Then the first-order logical theory of $(\\mathbb{N}\,+\,X\,
Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi
that the first-order logical theory of $(\\mathbb{N}\,+\,X)$ is decidable.
Our work strengthens and depends on earlier work of Villemaire and Bès.\
n\nThe essence of Cobham's theorem is that recognizability depends strongl
y on the choice of the base $k$. Our results strengthens this: two non-Pre
sburger definable sets that are recognizable in multiplicatively independe
nt bases\, are not only distinct\, but together computationally intractabl
e over Presburger arithmetic.\n\nThis is joint work with Christian Schulz.
\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lulu Fang (Nanjing University of Science and Technology)
DTSTART:20211005T123000Z
DTEND:20211005T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/58
DESCRIPTION:Title: On upper and lower fast Khintchine spectra in continued fractions
\nby Lulu Fang (Nanjing University of Science and Technology) as part of O
ne World Numeration seminar\n\n\nAbstract\nLet $\\psi:\\mathbb{N}\\to \\ma
thbb{R}^+$ be a function satisfying $\\psi(n)/n\\to \\infty$ as $n \\to \\
infty$. \nWe investigate from a multifractal analysis point of view the gr
owth speed of the sums $\\sum^n_{k=1}\\log a_k(x)$ \nwith respect to $\\ps
i(n)$\, where $x=[a_1(x)\,a_2(x)\,\\cdots]$ denotes the continued fraction
expansion of $x\\in (0\,1)$. \nThe (upper\, lower) fast Khintchine spectr
um is defined as the Hausdorff dimension of the set of points $x\\in(0\,1)
$ \nfor which the (upper\, lower) limit of $\\frac{1}{\\psi(n)}\\sum^n_{k=
1}\\log a_k(x)$ is equal to $1$. These three spectra \nhave been studied b
y Fan\, Liao \,Wang \\& Wu (2013\, 2016)\, Liao \\& Rams (2016). In this t
alk\, we will give a new look \nat the fast Khintchine spectrum\, and prov
ide a full description of upper and lower fast Khintchine spectra. The lat
ter \nimproves a result of Liao and Rams (2016).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liangang Ma (Binzhou University)
DTSTART:20211012T123000Z
DTEND:20211012T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/59
DESCRIPTION:Title: Inflection points in the Lyapunov spectrum for IFS on intervals\n
by Liangang Ma (Binzhou University) as part of One World Numeration semina
r\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Taylor Jones (University of North Texas)
DTSTART:20211005T130000Z
DTEND:20211005T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/60
DESCRIPTION:Title: On the Existence of Numbers with Matching Continued Fraction and Deci
mal Expansion\nby Taylor Jones (University of North Texas) as part of
One World Numeration seminar\n\n\nAbstract\nA Trott number in base 10 is o
ne whose continued fraction expansion agrees with its base 10 expansion in
the sense that $[0\;a_1\,a_2\,\\dots] = 0.(a_1)(a_2) \\cdots$ where $(a_i
)$ represents the string of digits of $a_i$. As an example $[0\;3\,29\,54\
,7\,\\dots] = 0.329547\\cdots$.\nAn analogous definition may be given for
a Trott number in any integer base $b>1$\, the set of which we denote by $
T_b$. The first natural question is whether $T_b$ is empty\, and if not\,
for which $b$? We discuss the history of the problem\, and give a heurist
ic process for constructing such numbers. We show that $T_{10}$ is indeed
non-empty\, and uncountable. With more delicate techniques\, a complete cl
assification may be given to all $b$ for which $T_b$ is non-empty. We also
discuss some further results\, such as a (non-trivial) upper bound on the
Hausdorff dimension of $T_b$\, as well as the question of whether the int
ersection of $T_b$ and $T_c$ can be non-empty.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mélodie Lapointe (Université de Paris)
DTSTART:20211019T123000Z
DTEND:20211019T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/61
DESCRIPTION:Title: q-analog of the Markoff injectivity conjecture\nby Mélodie Lapoi
nte (Université de Paris) as part of One World Numeration seminar\n\n\nAb
stract\nThe Markoff injectivity conjecture states that $w\\mapsto\\mu(w)_{
12}$ is injective on the set of Christoffel words where $\\mu:\\{\\mathtt{
0}\,\\mathtt{1}\\}^*\\to\\mathrm{SL}_2(\\mathbb{Z})$ is a certain homomorp
hism and $M_{12}$ is the entry above the diagonal of a $2\\times2$ matrix
$M$. Recently\, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\\
mu_q$ of $\\mu$ such that $\\mu_{q\\to1}(w)_{12}=\\mu(w)_{12}$ is the Mark
off number associated to the Christoffel word $w$. We show that there exis
ts an order $<_{radix}$ on $\\{\\mathtt{0}\,\\mathtt{1}\\}^*$ such that fo
r every balanced sequence $s \\in \\{\\mathtt{0}\,\\mathtt{1}\\}^\\mathbb{
Z}$ and for all factors $u\, v$ in the language of $s$ with $u <_{radix} v
$\, the difference $\\mu_q(v)_{12} - \\mu_q(u)_{12}$ is a nonzero polynomi
al of indeterminate $q$ with nonnegative integer coefficients. Therefore\,
for every $q>0$\, the map $\\{\\mathtt{0}\,\\mathtt{1}\\}^*\\to\\mathbb{R
}$ defined by $w\\mapsto\\mu_q(w)_{12}$ is increasing thus injective over
the language of a balanced sequence. The proof uses an equivalence betwe
en balanced sequences satisfying some Markoff property and indistinguishab
le asymptotic pairs.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Baake (Universität Bielefeld)
DTSTART:20211026T123000Z
DTEND:20211026T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/62
DESCRIPTION:Title: Spectral aspects of aperiodic dynamical systems\nby Michael Baake
(Universität Bielefeld) as part of One World Numeration seminar\n\n\nAbs
tract\nOne way to analyse aperiodic systems employs spectral notions\, eit
her via dynamical systems theory or via harmonic analysis. In this talk\,
we will look at two particular aspects of this\, after a quick overview of
how the diffraction measure can be used for this purpose. First\, we cons
ider some concequences of inflation rules on the spectra via renormalisati
on\, and how to use it to exclude absolutely continuous componenta. Second
\, we take a look at a class of dynamical systems of number-theoretic orig
in\, how they fit into the spectral picture\, and what (other) methods the
re are to distinguish them.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pieter Allaart (University of North Texas)
DTSTART:20211102T133000Z
DTEND:20211102T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/63
DESCRIPTION:Title: On the existence of Trott numbers relative to multiple bases\nby
Pieter Allaart (University of North Texas) as part of One World Numeration
seminar\n\n\nAbstract\nTrott numbers are real numbers in the interval $(0
\,1)$ whose continued fraction expansion equals their base-$b$ expansion\,
in a certain liberal but natural sense. They exist in some bases\, but no
t in all. In a previous OWNS talk\, T. Jones sketched a proof of the exist
ence of Trott numbers in base 10. In this talk I will discuss some further
properties of these Trott numbers\, and focus on the question: Can a numb
er ever be Trott in more than one base at once? While the answer is almost
certainly "no"\, a full proof of this seems currently out of reach. But w
e obtain some interesting partial answers by using a deep theorem from Dio
phantine approximation.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiqiang Wang (East China Normal University)
DTSTART:20211109T133000Z
DTEND:20211109T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/64
DESCRIPTION:Title: How inhomogeneous Cantor sets can pass a point\nby Zhiqiang Wang
(East China Normal University) as part of One World Numeration seminar\n\n
\nAbstract\nAbstract: For $x > 0$\, we define $$\\Upsilon(x) = \\{ (a\,b):
x\\in E_{a\,b}\, a>0\, b>0\, a+b \\le 1 \\}\,$$ where the set $E_{a\,b}$
is the unique nonempty compact invariant set generated by the inhomogeneou
s IFS $$\\{ f_0(x) = a x\, f_1(x) = b(x+1) \\}.$$ We show the set $\\Upsi
lon(x)$ is a Lebesgue null set with full Hausdorff dimension in $\\mathbb{
R}^2$\, and the intersection of sets $\\Upsilon(x_1)\, \\Upsilon(x_2)\, \\
dots\, \\Upsilon(x_\\ell)$ still has full Hausdorff dimension $\\mathbb{R}
^2$ for any finitely many positive real numbers $x_1\, x_2\, \\dots\, x_\\
ell$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucía Rossi (Montanuniversität Leoben)
DTSTART:20211116T133000Z
DTEND:20211116T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/65
DESCRIPTION:Title: Rational self-affine tiles associated to (nonstandard) digit systems<
/a>\nby Lucía Rossi (Montanuniversität Leoben) as part of One World Nume
ration seminar\n\n\nAbstract\nIn this talk we will introduce the notion of
rational self-affine tiles\, which are fractal-like sets that arise as th
e solution of a set equation associated to a digit system that consists of
a base\, given by an expanding rational matrix\, and a digit set\, given
by vectors. They can be interpreted as the set of “fractional parts” o
f this digit system\, and the challenge of this theory is that these sets
do not live in a Euclidean space\, but on more general spaces defined in t
erms of Laurent series. Steiner and Thuswaldner defined rational self-affi
ne tiles for the case where the base is a rational matrix with irreducible
characteristic polynomial. We present some tiling results that generalize
the ones obtained by Lagarias and Wang: we consider arbitrary expanding r
ational matrices as bases\, and simultaneously allow the digit sets to be
nonstandard (meaning they are not a complete set of residues modulo the ba
se). We also state some topological properties of rational self-affine til
es and give a criterion to guarantee positive measure in terms of the digi
t set.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sascha Troscheit (Universität Wien)
DTSTART:20211123T133000Z
DTEND:20211123T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/66
DESCRIPTION:Title: Analogues of Khintchine's theorem for random attractors\nby Sasch
a Troscheit (Universität Wien) as part of One World Numeration seminar\n\
n\nAbstract\nKhintchine’s theorem is an important result in number theor
y which links the Lebesgue measure of certain limsup sets with the converg
ence/divergence of naturally occurring volume sums. This behaviour has bee
n observed for deterministic fractal sets and inspired by this we investig
ate the random settings. Introducing randomisation into the problem makes
some parts more tractable\, while posing separate new challenges. In this
talk\, I will present joint work with Simon Baker where we provide suffici
ent conditions for a large class of stochastically self-similar and self-a
ffine attractors to have positive Lebesgue measure.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jamie Walton (University of Glasgow)
DTSTART:20211207T133000Z
DTEND:20211207T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/67
DESCRIPTION:Title: Extending the theory of symbolic substitutions to compact alphabets\nby Jamie Walton (University of Glasgow) as part of One World Numeratio
n seminar\n\n\nAbstract\nIn this work\, joint with Neil Mañibo and Dan Ru
st\, we consider an extension of the theory of symbolic substitutions to i
nfinite alphabets\, by requiring the alphabet to carry a compact\, Hausdor
ff topology for which the substitution is continuous. Such substitutions h
ave been considered before\, in particular by Durand\, Ormes and Petite fo
r zero-dimensional alphabets\, and Queffélec in the constant length case.
We find a simple condition which ensures that an associated substitution
operator is quasi-compact\, which we conjecture to always be satisfied for
primitive substitutions on countable alphabets. In the primitive case thi
s implies the existence of a unique natural tile length function and\, for
a recognisable substitution\, that the associated shift space is uniquely
ergodic. The main tools come from the theory of positive operators on Ban
ach spaces. Very few prerequisites will be assumed\, and the theory will b
e demonstrated via examples.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Younès Tierce (Université de Rouen Normandie)
DTSTART:20211109T140000Z
DTEND:20211109T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/68
DESCRIPTION:Title: Extensions of the random beta-transformation\nby Younès Tierce (
Université de Rouen Normandie) as part of One World Numeration seminar\n\
n\nAbstract\nLet $\\beta \\in (1\,2)$ and $I_\\beta := [0\,\\frac{1}{\\bet
a-1}]$. Almost every real number of $I_\\beta$ has infinitely many expansi
ons in base $\\beta$\, and the random $\\beta$-transformation generates al
l these expansions. We present the construction of a "geometrico-symbolic"
extension of the random $\\beta$-transformation\, providing a new proof o
f the existence and unicity of an absolutely continuous invariant probabil
ity measure\, and an expression of the density of this measure. This exten
sion shows off some nice renewal times\, and we use these to prove that th
e natural extension of the system is a Bernoulli automorphism.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valérie Berthé\, Pierre Arnoux\, ...
DTSTART:20211214T130000Z
DTEND:20211214T150000Z
DTSTAMP:20260404T131146Z
UID:OWNS/69
DESCRIPTION:Title: Special session commemorating Shunji Ito (1943-2021)\nby Valérie
Berthé\, Pierre Arnoux\, ... as part of One World Numeration seminar\n\n
\nAbstract\nIntroduction by Pierre Arnoux\, short talk by Valérie Berthé
\, contributions by Maki Furukado\, Cor Kraaikamp\, Hui Rao\, Robbie Robin
son\, Shin'Ichi Yasutomi\, Shigeki Akiyama\, and Hiromi Ei.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fan Lü (Sichuan Normal University)
DTSTART:20211221T133000Z
DTEND:20211221T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/70
DESCRIPTION:Title: Multiplicative Diophantine approximation in the parameter space of be
ta-dynamical system\nby Fan Lü (Sichuan Normal University) as part of
One World Numeration seminar\n\n\nAbstract\nBeta-transformation is a spec
ial kind of expanding dynamics\, the total information of which can be det
ermined by the orbits of some critical points (e.g.\, the point 1). Lettin
g $T_{\\beta}$ be the beta-transformation with $\\beta>1$ and $x$ be a fix
ed point in $(0\,1]$\, we consider the set of parameters $(\\alpha\, \\bet
a)$\, such that the multiple $\\|T^n_{\\alpha}(x)\\|\\|T^n_{\\beta}(x)\\|$
is well approximated or badly approximated. The Gallagher-type question\,
Jarník-type question as well as the badly approximable pairs\, i.e.\, Li
ttlewood-type question are studied in detail.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Gohlke (Universität Bielefeld)
DTSTART:20220111T133000Z
DTEND:20220111T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/71
DESCRIPTION:Title: Zero measure spectrum for multi-frequency Schrödinger operators\
nby Philipp Gohlke (Universität Bielefeld) as part of One World Numeratio
n seminar\n\n\nAbstract\nCantor spectrum of zero Lebesgue measure is a str
iking feature of Schrödinger operators associated with certain models of
aperiodic order\, like primitive substitution systems or Sturmian subshift
s. This is known to follow from a condition introduced by Boshernitzan tha
t establishes that on infinitely many scales words of the same length appe
ar with a similar frequency. Building on works of Berthé–Steiner–Thus
waldner and Fogg–Nous we show that on the two-dimensional torus\, Lebesg
ue almost every translation admits a natural coding such that the associat
ed subshift satisfies the Boshernitzan criterion (joint work with J.Chaika
\, D.Damanik and J.Fillman).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Agamemnon Zafeiropoulos (NTNU)
DTSTART:20220118T133000Z
DTEND:20220118T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/72
DESCRIPTION:Title: The order of magnitude of Sudler products\nby Agamemnon Zafeiropo
ulos (NTNU) as part of One World Numeration seminar\n\n\nAbstract\nGiven a
n irrational $\\alpha \\in [0\,1] \\smallsetminus \\mathbb{Q}$\, we define
the corresponding Sudler product by $$ P_N(\\alpha) = \\prod_{n=1}^{N}2|\
\sin (\\pi n \\alpha)|. $$ In joint work with C. Aistleitner and N. Techna
u\, we show that when $\\alpha = [0\;b\,b\,b…]$ is a quadratic irrationa
l with all partial quotients in its continued fraction expansion equal to
some integer b\, the following hold: \n\n- If $b\\leq 5$\, then $\\liminf_
{N\\to \\infty}P_N(\\alpha) >0$ and $\\limsup_{N\\to \\infty} P_N(\\alpha)
/N < \\infty$. \n\n-If $b\\geq 6$\, then $\\liminf_{N\\to \\infty}P_N(\\al
pha) = 0$ and $\\limsup_{N\\to \\infty} P_N(\\alpha)/N = \\infty$. \n\nWe
also present an analogue of the previous result for arbitrary quadratic ir
rationals (joint work with S. Grepstad and M. Neumueller).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Bonanno (Università di Pisa)
DTSTART:20220125T133000Z
DTEND:20220125T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/73
DESCRIPTION:Title: Infinite ergodic theory and a tree of rational pairs\nby Claudio
Bonanno (Università di Pisa) as part of One World Numeration seminar\n\n\
nAbstract\nThe study of the continued fraction expansions of real numbers
by ergodic methods is now a classical and well-known part of the theory of
dynamical systems. Less is known for the multi-dimensional expansions. I
will present an ergodic approach to a two-dimensional continued fraction a
lgorithm introduced by T. Garrity\, and show how to get a complete tree of
rational pairs by using the Farey sum of fractions. The talk is based on
joint work with A. Del Vigna and S. Munday.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Magdaléna Tinková (Czech Technical University in Prague)
DTSTART:20220208T133000Z
DTEND:20220208T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/74
DESCRIPTION:Title: Universal quadratic forms\, small norms and traces in families of num
ber fields\nby Magdaléna Tinková (Czech Technical University in Prag
ue) as part of One World Numeration seminar\n\n\nAbstract\nIn this talk\,
we will discuss universal quadratic forms over number fields and their con
nection with additively indecomposable integers. In particular\, we will f
ocus on Shanks' family of the simplest cubic fields. This is joint work wi
th Vítězslav Kala.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Coons (Universität Bielefeld)
DTSTART:20220308T133000Z
DTEND:20220308T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/75
DESCRIPTION:Title: A spectral theory of regular sequences\nby Michael Coons (Univers
ität Bielefeld) as part of One World Numeration seminar\n\n\nAbstract\nA
few years ago\, Michael Baake and I introduced a probability measure assoc
iated to Stern’s diatomic sequence\, an example of a regular sequence—
sequences which generalise constant length substitutions to infinite alpha
bets. In this talk\, I will discuss extensions of these results to more ge
neral regular sequences as well as further properties of these measures. T
his is joint work with several people\, including Michael Baake\, James Ev
ans\, Zachary Groth and Neil Manibo.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonas Jankauskas (Vilnius University)
DTSTART:20220201T133000Z
DTEND:20220201T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/76
DESCRIPTION:Title: Digit systems with rational base matrix over lattices\nby Jonas J
ankauskas (Vilnius University) as part of One World Numeration seminar\n\n
\nAbstract\nLet $A$ be a matrix with rational entries and no eigenvalue in
absolute value smaller than 1. Let $\\mathbb{Z}^d[A]$ be the minimal $A$-
invariant $\\mathbb{Z}$-module\, generated by integer vectors and the matr
ix $A$. In 2018\, we have shown that one can find a finite set $D$ of vect
ors\, such that each element of $\\mathbb{Z}^d[A]$ has a finite radix expa
nsion in base $A$ using only the digits from $D$\, i.e. $\\mathbb{Z}^d[A]=
D[A]$. This is called 'the finiteness property' of a digit system. In the
present talk I will review more recent developments in mathematical machin
ery\, that enable us to build finite digit systems over lattices using rea
sonably small digit sets\, and even to do some practical computations with
them on a computer. Tools that we use are the generalized rotation bases
with digit sets that have 'good' convex properties\, the semi-direct ('twi
sted') sums of such rotational digit systems\, and the special\, 'restrict
ed' version of the remainder division that preserves the lattice $\\mathbb
{Z}^d$ and can be extended to $\\mathbb{Z}^d[A]$. This is joint work with
J. Thuswaldner\, "Rational Matrix Digit Systems"\, to appear in "Linear an
d Multilinear Algebra".\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Steiner (CNRS\, Université de Paris)
DTSTART:20220215T133000Z
DTEND:20220215T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/77
DESCRIPTION:Title: Unique double base expansions\nby Wolfgang Steiner (CNRS\, Univer
sité de Paris) as part of One World Numeration seminar\n\n\nAbstract\nFor
pairs of real bases $\\beta_0\, \\beta_1 > 1$\, we study expansions of th
e form\n$\\sum_{k=1}^\\infty i_k / (\\beta_{i_1} \\beta_{i_2} \\cdots \\be
ta_{i_k})$\nwith digits $i_k \\in \\{0\,1\\}$.\nWe characterise the pairs
admitting non-trivial unique expansions as well as those admitting uncount
ably many unique expansions\, extending recent results of Neunhäuserer (2
021) and Zou\, Komornik and Lu (2021).\nSimilarly to the study of unique $
\\beta$-expansions with three digits by the speaker (2020)\, this boils do
wn to determining the cardinality of binary shifts defined by lexicographi
c inequalities.\nLabarca and Moreira (2006) characterised when such a shif
t is empty\, at most countable or uncountable\, depending on the position
of the lower and upper bounds with respect to Thue-Morse-Sturmian words. \
n\nThis is joint work with Vilmos Komornik and Yuru Zou.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Krenn (Universität Salzburg)
DTSTART:20220301T133000Z
DTEND:20220301T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/78
DESCRIPTION:Title: $k$-regular sequences: Asymptotics and Decidability\nby Daniel Kr
enn (Universität Salzburg) as part of One World Numeration seminar\n\n\nA
bstract\nA sequence $x(n)$ is called $k$-regular\, if the set of subsequen
ces $x(k^j n + r)$ is contained in a finitely generated module. In this ta
lk\, we will consider the asymptotic growth of $k$-regular sequences. When
is it possible to compute it? ...and when not? If possible\, how precisel
y can we compute it? If not\, is it just a lack of methods or are the unde
rlying decision questions recursively solvable (i.e.\, decidable in a comp
utational sense)? We will discuss answers to these questions. To round off
the picture\, we will consider further decidability questions around $k$-
regular sequences and the subclass of $k$-automatic sequences.\n\nThis is
based on joint works with Clemens Heuberger and with Jeffrey Shallit.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Popoli (Université de Lorraine)
DTSTART:20220315T133000Z
DTEND:20220315T143000Z
DTSTAMP:20260404T131146Z
UID:OWNS/79
DESCRIPTION:Title: Maximum order complexity for some automatic and morphic sequences alo
ng polynomial values\nby Pierre Popoli (Université de Lorraine) as pa
rt of One World Numeration seminar\n\n\nAbstract\nAutomatic sequences are
not suitable sequences for cryptographic applications since both their sub
word complexity and their expansion complexity are small\, and their corre
lation measure of order 2 is large. These sequences are highly predictable
despite having a large maximum order complexity. However\, recent results
show that polynomial subsequences of automatic sequences\, such as the Th
ue-Morse sequence or the Rudin-Shapiro sequence\, are better candidates fo
r pseudorandom sequences. A natural generalization of automatic sequences
are morphic sequences\, given by a fixed point of a prolongeable morphism
that is not necessarily uniform. In this talk\, I will present my results
on lowers bounds for the maximum order complexity of the Thue-Morse sequen
ce\, the Rudin-Shapiro sequence and the sum of digits function in Zeckendo
rf base\, which are respectively automatics and morphic sequences.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tingyu Zhang (East China Normal University)
DTSTART:20220329T123000Z
DTEND:20220329T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/80
DESCRIPTION:Title: Random $\\beta$-transformation on fat Sierpiński gasket\nby Ting
yu Zhang (East China Normal University) as part of One World Numeration se
minar\n\n\nAbstract\nWe define the notions of greedy\, lazy and random tra
nsformations on fat Sierpiński gasket. We determine the bases\, for which
the system has a unique measure of maximal entropy and an invariant measu
re of product type\, with one coordinate being absolutely continuous with
respect to Lebesgue measure. \n\nThis is joint work with K. Dajani and W.
Li.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungwon Lee (University of Warwick)
DTSTART:20220405T123000Z
DTEND:20220405T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/81
DESCRIPTION:Title: Dynamics of Ostrowski skew-product: Limit laws and Hausdorff dimensio
ns\nby Jungwon Lee (University of Warwick) as part of One World Numera
tion seminar\n\n\nAbstract\nWe discuss a dynamical study of the Ostrowski
skew-product map in the context of inhomogeneous Diophantine approximation
. We plan to outline the setup/ strategy based on transfer operator analys
is and applications in arithmetic of number fields (joint with Valérie Be
rthé).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET)
DTSTART:20220412T123000Z
DTEND:20220412T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/82
DESCRIPTION:Title: Lochs-type theorems beyond positive entropy\nby Eda Cesaratto (Un
iv. Nac. de Gral. Sarmiento & CONICET) as part of One World Numeration sem
inar\n\n\nAbstract\nLochs' theorem and its generalizations are conversion
theorems that relate the number of digits determined in one expansion of a
real number as a function of the number of digits given in some other exp
ansion. In its original version\, Lochs' theorem related decimal expansion
s with continued fraction expansions. Such conversion results can also be
stated for sequences of interval partitions under suitable assumptions\, w
ith results holding almost everywhere\, or in measure\, involving the entr
opy. This is the viewpoint we develop here. In order to deal with sequence
s of partitions beyond positive entropy\, this paper introduces the notion
of log-balanced sequences of partitions\, together with their weight func
tions. These are sequences of interval partitions such that the logarithms
of the measures of their intervals at each depth are roughly the same. We
then state Lochs-type theorems which work even in the case of zero entrop
y\, in particular for several important log-balanced sequences of partitio
ns of a number-theoretic nature. \n\nThis is joint work with Valérie Bert
hé (IRIF)\, Pablo Rotondo (U. Gustave Eiffel) and Martín Safe (Univ. Nac
. del Sur & CONICET\, Argentina).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paulina Cecchi Bernales (Universidad de Chile)
DTSTART:20220419T123000Z
DTEND:20220419T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/83
DESCRIPTION:Title: Coboundaries and eigenvalues of finitary S-adic systems\nby Pauli
na Cecchi Bernales (Universidad de Chile) as part of One World Numeration
seminar\n\n\nAbstract\nAn S-adic system is a shift space obtained by perfo
rming an infinite composition of morphisms defined over possibly different
finite alphabets. It is said to be finitary if these morphisms are taken
from a finite set. S-adic systems are a generalization of substitution shi
fts. In this talk we will discuss spectral properties of finitary S-adic s
ystems. Our departure point will be a theorem by B. Host which characteriz
es eigenvalues of substitution shifts\, and where coboundaries appear as a
key tool. We will introduce the notion of S-adic coboundaries and present
some results which show how they are related with eigenvalues of S-adic s
ystems. We will also present some applications of our results to constant-
length finitary S-adic systems. \n\nThis is joint work with Valérie Berth
é and Reem Yassawi.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Chevallier (Université de Haute Alsace)
DTSTART:20220503T123000Z
DTEND:20220503T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/84
DESCRIPTION:Title: Best Diophantine approximations in the complex plane with Gaussian in
tegers\nby Nicolas Chevallier (Université de Haute Alsace) as part of
One World Numeration seminar\n\n\nAbstract\nStarting with the minimal vec
tors in lattices over Gaussian integers in $\\C^2$\, we define a algorithm
that finds the sequence of minimal vectors of any unimodular lattice in $
\\C^2$.\nRestricted to lattices associated with complex numbers this algor
ithm find all the best Diophantine approximations of a complex numbers.\nF
ollowing Doeblin\, Lenstra\, Bosma\, Jager and Wiedijk\, we study the limi
t distribution of the sequence of products $(u_{n1}u_{n2})_n$ where $(u _n
=( u_{n1}\,u_{n2} ))_n$ is the sequence of minimal vectors of a lattice in
$C^2$. We show that there exists a measure in $\\C$ which is the limit di
stribution of the sequence of products of almost all unimodular lattices.
\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vilmos Komornik (Shenzhen University and Université de Strasbourg
)
DTSTART:20220517T123000Z
DTEND:20220517T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/85
DESCRIPTION:Title: Topology of univoque sets in real base expansions\nby Vilmos Komo
rnik (Shenzhen University and Université de Strasbourg) as part of One Wo
rld Numeration seminar\n\n\nAbstract\nWe report on a recent joint paper wi
th Martijn de Vries and Paola Loreti. Given a positive integer $M$ and a r
eal number $1 < q\\le M+1$\, an expansion of a real number $x \\in \\left[
0\,M/(q-1)\\right]$ over the alphabet $A=\\{0\,1\,\\ldots\,M\\}$ is a sequ
ence $(c_i) \\in A^{\\mathbb{N}}$ such that $x=\\sum_{i=1}^{\\infty}c_iq^{
-i}$. Generalizing many earlier results\, we investigate the topological p
roperties of the set $U_q$ consisting of numbers $x$ having a unique expan
sion of this form\, and the combinatorial properties of the set $U_q'$ con
sisting of their corresponding expansions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Émilie Charlier (Université de Liège)
DTSTART:20220524T123000Z
DTEND:20220524T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/86
DESCRIPTION:Title: Spectrum\, algebraicity and normalization in alternate bases\nby
Émilie Charlier (Université de Liège) as part of One World Numeration s
eminar\n\n\nAbstract\nThe first aim of this work is to give information ab
out the algebraic properties of alternate bases determining sofic systems.
We exhibit two conditions: one necessary and one sufficient. Comparing th
e setting of alternate bases to that of one real base\, these conditions e
xhibit a new phenomenon: the bases should be expressible as rational funct
ions of their product. The second aim is to provide an analogue of Frougny
's result concerning normalization of real bases representations. Under so
me suitable condition (i.e.\, our previous sufficient condition for being
a sofic system)\, we prove that the normalization function is computable b
y a finite Büchi automaton\, and furthermore\, we effectively construct s
uch an automaton. An important tool in our study is the spectrum of numera
tion systems associated with alternate bases. For our purposes\, we use a
generalized concept of spectrum associated with a complex base and complex
digits\, and we study its topological properties. \n\nThis is joint work
with Célia Cisternino\, Zuzana Masáková and Edita Pelantová.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verónica Becher (Universidad de Buenos Aires & CONICET Argentina)
DTSTART:20220531T123000Z
DTEND:20220531T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/87
DESCRIPTION:Title: Poisson generic real numbers\nby Verónica Becher (Universidad de
Buenos Aires & CONICET Argentina) as part of One World Numeration seminar
\n\n\nAbstract\nYears ago Zeev Rudnick defined the Poisson generic real nu
mbers as those where the number of occurrences of the long strings in the
initial segments of their fractional expansions in some base have the Poi
sson distribution. Yuval Peres and Benjamin Weiss proved that almost all r
eal numbers\, with respect to Lebesgue measure\, are Poisson generic. They
also showed that Poisson genericity implies Borel normality but the two n
otions do not coincide\, witnessed by the famous Champernowne constant.
We recently showed that there are computable Poisson generic real numbers
and that all Martin-Löf real numbers are Poisson generic. \nThis is join
t work Nicolás Álvarez and Martín Mereb.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sophie Morier-Genoud (Université Reims Champagne Ardenne)
DTSTART:20220607T123000Z
DTEND:20220607T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/88
DESCRIPTION:Title: q-analogues of real numbers\nby Sophie Morier-Genoud (Université
Reims Champagne Ardenne) as part of One World Numeration seminar\n\n\nAbs
tract\nClassical sequences of numbers often lead to interesting q-analogue
s. The most popular among them are certainly the q-integers and the q-bino
mial coefficients which both appear in various areas of mathematics and ph
ysics. With Valentin Ovsienko we recently suggested a notion of q-rational
s based on combinatorial properties and continued fraction expansions. The
definition of q-rationals naturally extends the one of q-integers and lea
ds to a ratio of polynomials with positive integer coefficients. I will ex
plain the construction and give the main properties. In particular I will
briefly mention connections with the combinatorics of posets\, cluster alg
ebras\, Jones polynomials\, homological algebra. Finally I will also prese
nt further developments of the theory\, leading to the notion of q-irratio
nals and q-unimodular matrices.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James A. Yorke (University of Maryland)
DTSTART:20220621T123000Z
DTEND:20220621T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/89
DESCRIPTION:Title: Large and Small Chaos Models\nby James A. Yorke (University of Ma
ryland) as part of One World Numeration seminar\n\n\nAbstract\nTo set the
scene\, I will discuss one large model\, a whole-Earth model for predictin
g the weather\, and how to initialize such a model and what aspects of cha
os are essential. Then I will discuss a couple related “very simple” m
aps that tell us a great deal about very complex models. The results on si
mple models are new. I will discuss the logistic map mx(1-x). Its dynamics
can make us rethink climate models. Also\, we have created a piecewise li
near map on a 3D cube that is unstable in 2 dimensions in some places and
unstable in 1 in others. It has a dense set of periodic points that are 1
D unstable and another dense set of periodic points that are all 2 D unsta
ble. I will also discuss a new project whose tentative title is “ Can th
e flap of butterfly's wings shift a tornado out of Texas -- without chaos?
\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlene Kalle (Universiteit Leiden)
DTSTART:20220705T123000Z
DTEND:20220705T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/90
DESCRIPTION:Title: Random Lüroth expansions\nby Charlene Kalle (Universiteit Leiden
) as part of One World Numeration seminar\n\n\nAbstract\nSince the introdu
ction of Lüroth expansions by Lüroth in his paper from 1883 many results
have appeared on their approximation properties. In 1990 Kalpazidou\, Kno
pfmacher and Knopfmacher introduced alternating Lüroth expansions and stu
died their properties. A comparison between the two and other comparable n
umber systems was then given by Barrionuevo\, Burton\, Dajani and Kraaikam
p in 1996. In this talk we introduce a family of random dynamical systems
that produce many Lüroth type expansions at once. Topics that we consider
are periodic expansions\, universal expansions\, speed of convergence and
approximation coefficients. This talk is based on joint work with Marta M
aggioni.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruofan Li (South China University of Technology)
DTSTART:20220712T123000Z
DTEND:20220712T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/91
DESCRIPTION:Title: Rational numbers in $\\times b$-invariant sets\nby Ruofan Li (Sou
th China University of Technology) as part of One World Numeration seminar
\n\n\nAbstract\nLet $b \\ge 2$ be an integer and $S$ be a finite non-empty
set of primes not containing divisors of $b$. For any $\\times b$-invaria
nt\, non-dense subset $A$ of $[0\,1)$\, we prove the finiteness of rationa
l numbers in $A$ whose denominators can only be divided by primes in $S$.
A quantitative result on the largest prime divisors of the denominators of
rational numbers in $A$ is also obtained. \nThis is joint work with Bing
Li and Yufeng Wu.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedict Sewell (Alfréd Rényi Institute)
DTSTART:20220913T123000Z
DTEND:20220913T133000Z
DTSTAMP:20260404T131146Z
UID:OWNS/92
DESCRIPTION:Title: An upper bound on the box-counting dimension of the Rauzy gasket\
nby Benedict Sewell (Alfréd Rényi Institute) as part of One World Numera
tion seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niels Langeveld (Montanuniversität Leoben)
DTSTART:20220927T120000Z
DTEND:20220927T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/93
DESCRIPTION:Title: $N$-continued fractions and $S$-adic sequences\nby Niels Langevel
d (Montanuniversität Leoben) as part of One World Numeration seminar\n\n\
nAbstract\nGiven the $N$-continued fraction of a number $x$\, we construct
$N$-continued fraction sequences in the same spirit as Sturmian sequences
can be constructed from regular continued fractions. These sequences are
infinite words over a two letter alphabet obtained as the limit of a direc
tive sequence of certain substitutions (they are S-adic sequences). By vie
wing them as a generalisation of Sturmian sequences it is natural to study
balancedness. We will see that the sequences we construct are not 1-balan
ced but C-balanced for $C=N^2$. Furthermore\, we construct a dual sequence
which is related to the natural extension of the $N$-continued fraction a
lgorithm. This talk is joint work with Lucía Rossi and Jörg Thuswaldner.
\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Siukaev (Higher School of Economics)
DTSTART:20221004T120000Z
DTEND:20221004T123000Z
DTSTAMP:20260404T131146Z
UID:OWNS/94
DESCRIPTION:Title: Exactness and ergodicity of certain Markovian multidimensional fracti
on algorithms\nby David Siukaev (Higher School of Economics) as part o
f One World Numeration seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lukas Spiegelhofer (Montanuniversität Leoben)
DTSTART:20221011T120000Z
DTEND:20221011T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/95
DESCRIPTION:Title: Primes as sums of Fibonacci numbers\nby Lukas Spiegelhofer (Monta
nuniversität Leoben) as part of One World Numeration seminar\n\n\nAbstrac
t\nWe prove that the Zeckendorf sum-of-digits function of prime numbers\,
$z(p)$\, is uniformly distributed in residue classes.\nThe main ingredient
that made this proof possible is the study of very sparse arithmetic subs
equences of $z(n)$. In other words\, we will meet the level of distributio
n.\nOur proof of this central result is based on a combination of the "Mau
duit−Rivat−van der Corput method" for digital problems and an estimate
of a Gowers norm related to $z(n)$.\nOur method of proof yields examples
of substitutive sequences that are orthogonal to the Möbius function (cf.
Sarnak's conjecture).\n\nThis is joint work with Michael Drmota and Cleme
ns Müllner (TU Wien).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Skripchenko (Higher School of Economics)
DTSTART:20221004T123000Z
DTEND:20221004T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/96
DESCRIPTION:Title: Bruin-Troubetzkoy family of interval translation mappings: a new glan
ce\nby Alexandra Skripchenko (Higher School of Economics) as part of O
ne World Numeration seminar\n\n\nAbstract\nIn 2002 H. Bruin and S. Troubet
zkoy described a special class of interval translation mappings on three i
ntervals. They showed that in this class the typical ITM could be reduced
to an interval exchange transformations. They also proved that generic ITM
of their class that can not be reduced to IET is uniquely ergodic. \n\nWe
suggest an alternative proof of the first statement and get a stronger ve
rsion of the second one. It is a joint work in progress with Mauro Artigia
ni and Pascal Hubert.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Álvaro Bustos-Gajardo (The Open University)
DTSTART:20221025T120000Z
DTEND:20221025T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/97
DESCRIPTION:Title: Quasi-recognizability and continuous eigenvalues of torsion-free S-ad
ic systems\nby Álvaro Bustos-Gajardo (The Open University) as part of
One World Numeration seminar\n\n\nAbstract\nWe discuss combinatorial and
dynamical descriptions of S-adic systems generated by sequences of constan
t-length morphisms between alphabets of bounded size. For this purpose\, w
e introduce the notion of quasi-recognisability\, a strictly weaker versio
n of recognisability but which is indeed enough to reconstruct several cla
ssical arguments of the theory of constant-length substitutions in this mo
re general context. Furthermore\, we identify a large family of directive
sequences\, which we call "torsion-free"\, for which quasi-recognisability
is obtained naturally\, and can be improved to actual recognisability wit
h relative ease.\n\nUsing these notions we give S-adic analogues of the no
tions of column number and height for substitutions\, including dynamical
and combinatorial interpretations of each\, and give a general characteris
ation of the maximal equicontinuous factor of the identified family of S-a
dic shifts\, showing as a consequence that in this context all continuous
eigenvalues must be rational. As well\, we employ the tools developed for
a first approach to the measurable case.\n\nThis is a joint work with Neil
Mañibo and Reem Yassawi.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wen Wu (South China University of Technology)
DTSTART:20221108T130000Z
DTEND:20221108T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/98
DESCRIPTION:Title: From the Thue-Morse sequence to the apwenian sequences\nby Wen Wu
(South China University of Technology) as part of One World Numeration se
minar\n\n\nAbstract\nIn this talk\, we will introduce a class of $\\pm 1$
sequences\, called the apwenian sequences. The Hankel determinants of the
se $\\pm1$ sequences share the same property as the Hankel determinants of
the Thue-Morse sequence found by Allouche\, Peyrière\, Wen and Wen in 19
98. In particular\, the Hankel determinants of apwenian sequences do not
vanish. This allows us to discuss the Diophantine property of the values o
f their generating functions at $1/b$ where $b\\geq 2$ is an integer. More
over\, the number of $\\pm 1$ apwenian sequences is given explicitly. Sim
ilar questions are also discussed for $0$-$1$ apwenian sequences. This ta
lk is based on joint work with Y.-J. Guo and G.-N. Han.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Faustin Adiceam (Université Paris-Est Créteil)
DTSTART:20221122T130000Z
DTEND:20221122T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/99
DESCRIPTION:Title: Badly approximable vectors and Littlewood-type problems\nby Faust
in Adiceam (Université Paris-Est Créteil) as part of One World Numeratio
n seminar\n\n\nAbstract\nBadly approximable vectors are fractal sets enjoy
ing rich Diophantine properties. In this respect\, they play a crucial rol
e in many problems well beyond Number Theory and Fractal Geometry (e.g.\,
in signal processing\, in mathematical physics and in convex geometry). \n
\nAfter outlining some of the latest developments in this very active area
of research\, we will take an interest in the Littlewood conjecture (c. 1
930) and in its variants which all admit a natural formulation in terms of
properties satisfied by badly approximable vectors. We will then show how
ideas emerging from the mathematical theory of quasicrystals\, from numer
ation systems and from the theory of aperiodic tilings have recently been
used to refute the so-called t-adic Littlewood conjecture. \n\nAll necessa
ry concepts will be defined in the talk. Joint with Fred Lunnon (Maynooth)
and Erez Nesharim (Technion\, Haifa).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yufei Chen (TU Delft)
DTSTART:20221018T120000Z
DTEND:20221018T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/100
DESCRIPTION:Title: Matching of orbits of certain $N$-expansions with a finite set of di
gits\nby Yufei Chen (TU Delft) as part of One World Numeration seminar
\n\n\nAbstract\nIn this talk we consider a class of continued fraction exp
ansions: the so-called $N$-expansions with a finite digit set\, where $N\\
geq 2$ is an integer. For $N$ fixed they are steered by a parameter $\\alp
ha\\in (0\,\\sqrt{N}-1]$. For $N=2$ an explicit interval $[A\,B]$ was dete
rmined\, such that for all $\\alpha\\in [A\,B]$ the entropy $h(T_{\\alpha}
)$ of the underlying Gauss-map $T_{\\alpha}$ is equal. In this paper we sh
ow that for all $N\\in\n$\, $N\\geq 2$\, such plateaux exist. In order to
show that the entropy is constant on such plateaux\, we obtain the underly
ing planar natural extension of the maps $T_{\\alpha}$\, the $T_{\\alpha}$
-invariant measure\, ergodicity\, and we show that for any two $\\alpha\,\
\alpha'$ from the same plateau\, the natural extensions are metrically iso
morphic\, and the isomorphism is given explicitly. The plateaux are found
by a property called matching.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seul Bee Lee (Institute for Basic Science)
DTSTART:20221115T130000Z
DTEND:20221115T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/101
DESCRIPTION:Title: Regularity properties of Brjuno functions associated with by-excess\
, odd and even continued fractions\nby Seul Bee Lee (Institute for Bas
ic Science) as part of One World Numeration seminar\n\n\nAbstract\nAn irra
tional number is called a Brjuno number if the sum of the series of $\\log
(q_{n+1})/q_n$ converges\, where $q_n$ is the denominator of the $n$-th pr
incipal convergent of the regular continued fraction. The importance of Br
juno numbers comes from the study of one variable analytic small divisor p
roblems. In 1988\, J.-C. Yoccoz introduced the Brjuno function which chara
cterizes the Brjuno numbers to estimate the size of Siegel disks. In this
talk\, we introduce Brjuno-type functions associated with by-excess\, odd
and even continued fractions with a number theoretical motivation. Then we
discuss the $L^p$ and the Hölder regularity properties of the difference
between the classical Brjuno function and the Brjuno-type functions. This
is joint work with Stefano Marmi.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Hauke (TU Graz)
DTSTART:20221129T130000Z
DTEND:20221129T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/102
DESCRIPTION:Title: The asymptotic behaviour of Sudler products\nby Manuel Hauke (TU
Graz) as part of One World Numeration seminar\n\n\nAbstract\nGiven an irr
ational number $\\alpha$\, we study the asymptotic behaviour of the Sudler
product defined by $P_N(\\alpha) = \\prod_{r=1}^N 2 \\lvert \\sin \\pi r
\\alpha \\rvert$\, which appears in many different areas of mathematics.\n
In this talk\, we explain the connection between the size of $P_N(\\alpha)
$ and the Ostrowski expansion of $N$ with respect to $\\alpha$.\nWe show t
hat $\\liminf_{N \\to \\infty} P_N(\\alpha) = 0$ and $\\limsup_{N \\to \\i
nfty} P_N(\\alpha)/N = \\infty$\, whenever the sequence of partial quotien
ts in the continued fraction expansion of $\\alpha$ exceeds $7$ infinitely
often\, and show that the value $7$ is optimal.\n\nFor Lebesgue-almost ev
ery $\\alpha$\, we can prove more: we show that for every non-decreasing f
unction $\\psi: (0\,\\infty) \\to (0\,\\infty)$ with $\\sum_{k=1}^{\\infty
} \\frac{1}{\\psi(k)} = \\infty$ and\n$\\liminf_{k \\to \\infty} \\psi(k)/
(k \\log k)$ sufficiently large\, the conditions $\\log P_N(\\alpha) \\leq
-\\psi(\\log N)$\, $\\log P_N(\\alpha) \\geq \\psi(\\log N)$ hold on sets
of upper density $1$ respectively $1/2$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Bandt (Universität Greifswald)
DTSTART:20221206T130000Z
DTEND:20221206T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/103
DESCRIPTION:Title: Automata generated topological spaces and self-affine tilings\nb
y Christoph Bandt (Universität Greifswald) as part of One World Numeratio
n seminar\n\n\nAbstract\nNumeration assigns symbolic sequences as addresse
s to points in a space X. There are points which get multiple addresses.
It is known that these identifications describe the topology of X and can
often be determined by an automaton. Here we define a corresponding class
of automata and discuss their properties and interesting examples. Vario
us open questions concern the realization of such automata by iterated fun
ctions and the uniqueness of such an implementation. Self-affine tiles for
m a simple class of examples.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiroki Takahasi (Keio University)
DTSTART:20221213T130000Z
DTEND:20221213T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/104
DESCRIPTION:Title: Distribution of cycles for one-dimensional random dynamical systems<
/a>\nby Hiroki Takahasi (Keio University) as part of One World Numeration
seminar\n\n\nAbstract\nWe consider an independently identically distribute
d random dynamical system generated by finitely many\, non-uniformly expan
ding Markov interval maps with a finite number of branches.\nAssuming a to
pologically mixing condition and the uniqueness of equilibrium state for t
he associated skew product map\, we establish a samplewise (quenched) almo
st-sure level-2 weighted equidistribution of "random cycles"\, with respec
t to a natural stationary measure as the periods of the cycles tend to inf
inity. This result implies an analogue of Bowen's theorem on periodic orbi
ts of topologically mixing Axiom A diffeomorphisms. \n\nThis talk is based
on the preprint arXiv:2108.05522. If time permits\, I will mention some f
uture perspectives in this project.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roswitha Hofer (JKU Linz)
DTSTART:20230110T130000Z
DTEND:20230110T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/105
DESCRIPTION:Title: Exact order of discrepancy of normal numbers\nby Roswitha Hofer
(JKU Linz) as part of One World Numeration seminar\n\n\nAbstract\nIn the t
alk we discuss some previous results on the discrepancy of normal numbers
and consider the still open question of Korobov: What is the best possible
order of discrepancy $D_N$ in $N$\, a sequence $(\\{b^n\\alpha\\})_{n\\ge
q 0}$\, $b\\geq 2\,\\in\\mathbb{N}$\, can have for some real number $\\alp
ha$? If $\\lim_{N\\to\\infty} D_N=0$ then $\\alpha$ in called normal in ba
se $b$. \n\nSo far the best upper bounds for $D_N$ for explicitly known no
rmal numbers in base $2$ are of the form $ND_N\\ll\\log^2 N$. The first ex
ample is due to Levin (1999)\, which was later generalized by Becher and C
arton (2019). In this talk we discuss the recent result in joint work with
Gerhard Larcher that guarantees $ND_N\\gg \\log^2 N$ for Levin's binary n
ormal number. So EITHER $ND_N\\ll \\log^2N$ is the best possible order for
$D_N$ in $N$ of a normal number OR there exist another example of a binar
y normal number with a better growth of $ND_N$ in $N$. The recent result f
or Levin's normal number might support the conjecture that $ND_N\\ll \\log
^2N$ is the best order for $D_N$ in $N$ a normal number can obtain.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Slade Sanderson (Utrecht University)
DTSTART:20230131T130000Z
DTEND:20230131T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/106
DESCRIPTION:Title: Matching for parameterised symmetric golden maps\nby Slade Sande
rson (Utrecht University) as part of One World Numeration seminar\n\n\nAbs
tract\nIn 2020\, Dajani and Kalle investigated invariant measures and freq
uencies of digits of signed binary expansions arising from a parameterised
family of piecewise linear interval maps of constant slope 2. Central to
their study was a property called ‘matching\,’ where the orbits of th
e left and right limits of discontinuity points agree after some finite nu
mber of steps. We obtain analogous results for a parameterised family of
‘symmetric golden maps’ of constant slope $\\beta$\, with $\\beta$ the
golden mean. Matching is again central to our methods\, though the dynam
ics of the symmetric golden maps are more delicate than the binary case.
We characterize the matching phenomenon in our setting\, present explicit
invariant measures and frequencies of digits of signed $\\beta$-expansions
\, and---time permitting---show further implications for a family of piece
wise linear maps which arise as jump transformations of the symmetric gold
en maps. \n\nJoint with Karma Dajani.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiko Kawamura (University of North Texas)
DTSTART:20230124T130000Z
DTEND:20230124T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/107
DESCRIPTION:Title: The partial derivative of Okamoto's functions with respect to the pa
rameter\nby Kiko Kawamura (University of North Texas) as part of One W
orld Numeration seminar\n\n\nAbstract\nOkamoto's functions were introduced
in 2005 as a one-parameter family of self-affine functions\, which are ex
pressed by ternary expansion of $x$ on the interval $[0\,1]$. By changing
the parameter\, one can produce interesting examples: Perkins' nowhere dif
ferentiable function\, Bourbaki-Katsuura function and Cantor's Devil's sta
ircase function. \n\nIn this talk\, we consider the partial derivative of
Okomoto's functions with respect to the parameter $a$. We place a signific
ant focus on $a = 1/3$ to describe the properties of a nowhere differentia
ble function $K(x)$ for which the set of points of infinite derivative pro
duces an example of a measure zero set with Hausdorff dimension 1.\n\nThis
is a joint work with T. Mathis and M.Paizanis (undergraduate students) an
d N.Dalaklis (graduate student). The talk is very accessible and includes
many computer graphics.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derong Kong (Chongqing University)
DTSTART:20230307T130000Z
DTEND:20230307T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/108
DESCRIPTION:Title: Critical values for the beta-transformation with a hole at 0\nby
Derong Kong (Chongqing University) as part of One World Numeration semina
r\n\n\nAbstract\nGiven $\\beta \\in (1\,2]$\, let $T$ be the $\\beta$-tran
sformation on the unit circle $[0\,1)$. For $t \\in [0\,1)$ let $K(t)$ be
the survivor set consisting of all $x$ whose orbit under $T$ never hits th
e open interval $(0\,t)$. Kalle et al. [ETDS\, 2020] proved that the Hausd
orff dimension function $\\dim K(t)$ is a non-increasing Devil's staircase
in $t$. So there exists a critical value such that $\\dim K(t)$ is vanish
ing when $t$ is passing through this critical value. In this paper we will
describe this critical value and analyze its interesting properties. Our
strategy to find the critical value depends on certain substitutions of Fa
rey words and a renormalization scheme from dynamical systems. This is joi
nt work with Pieter Allaart.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Demi Allen (University of Exeter)
DTSTART:20230321T130000Z
DTEND:20230321T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/109
DESCRIPTION:Title: Diophantine Approximation for systems of linear forms - some comment
s on inhomogeneity\, monotonicity\, and primitivity\nby Demi Allen (Un
iversity of Exeter) as part of One World Numeration seminar\n\n\nAbstract\
nDiophantine Approximation is a branch of Number Theory in which the centr
al theme is understanding how well real numbers can be approximated by rat
ionals. In the most classical setting\, a $\\psi$-well-approximable number
is one which can be approximated by rationals to a given degree of accura
cy specified by an approximating function $\\psi$. Khintchine's Theorem pr
ovides a beautiful characterisation of the Lebesgue measure of the set of
$\\psi$-well-approximable numbers and is one of the cornerstone results of
Diophantine Approximation. In this talk I will discuss the generalisation
of Khintchine's Theorem to the setting of approximation for systems of li
near forms. I will focus mainly on the topic of inhomogeneous approximatio
n for systems of linear forms. Time permitting\, I may also discuss approx
imation for systems of linear forms subject to certain primitivity constra
ints. This talk will be based on joint work with Felipe Ramirez (Wesleyan\
, US).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ale Jan Homburg (University of Amsterdam\, VU University Amsterdam
)
DTSTART:20230207T130000Z
DTEND:20230207T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/110
DESCRIPTION:Title: Iterated function systems of linear expanding and contracting maps o
n the unit interval\nby Ale Jan Homburg (University of Amsterdam\, VU
University Amsterdam) as part of One World Numeration seminar\n\n\nAbstrac
t\nWe analyze the two-point motions of iterated function systems on the un
it interval generated by expanding and contracting affine maps\, where the
expansion and contraction rates are determined by a pair $(M\,N)$ of inte
gers.\n\nThis dynamics depends on the Lyapunov exponent.\n\nFor a negative
Lyapunov exponent we establish synchronization\, meaning convergence of o
rbits with different initial points. For a vanishing Lyapunov exponent we
establish intermittency\, where orbits are close for a set of iterates of
full density\, but are intermittently apart. For a positive Lyapunov expon
ent we show the existence of an absolutely continuous stationary measure f
or the two-point dynamics and discuss its consequences.\n\nFor nonnegative
Lyapunov exponent and pairs $(M\,N)$ that are multiplicatively dependent
integers\, we provide explicit expressions for absolutely continuous stati
onary measures of the two-point dynamics. These stationary measures are in
finite $\\sigma$-finite measures in the case of zero Lyapunov exponent.\n\
nThis is joint work with Charlene Kalle.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yining Hu (Huazhong University of Science and Technology)
DTSTART:20230214T130000Z
DTEND:20230214T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/111
DESCRIPTION:Title: Algebraic automatic continued fractions in characteristic 2\nby
Yining Hu (Huazhong University of Science and Technology) as part of One W
orld Numeration seminar\n\n\nAbstract\nWe present two families of automati
c sequences that define algebraic continued fractions in characteristic $2
$. The period-doubling sequence belongs to the first family $\\mathcal{P}$
\; and its sum modulo $2$\, the Thue-Morse sequence\, belongs to the secon
d family $\\mathcal{G}$. The family $\\mathcal{G}$ contains all the itera
ted sums of sequences from the $\\mathcal{P}$ and more.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roland Zweimüller (University of Vienna)
DTSTART:20230328T120000Z
DTEND:20230328T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/112
DESCRIPTION:Title: Variations on a theme of Doeblin\nby Roland Zweimüller (Univers
ity of Vienna) as part of One World Numeration seminar\n\n\nAbstract\nStar
ting from Doeblin's observation on the Poissonian nature of occurrences of
large digits in typical continued fraction expansions\, I will outline so
me recent work on rare events in measure preserving systems (including spa
tiotemporal and local limit theorems) which\, in particular\, allows us to
refine Doeblin's statement in several ways. \n\n(Part of this is joint wo
rk with Max Auer.)\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Lukyanenko (George Mason University)
DTSTART:20230418T120000Z
DTEND:20230418T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/113
DESCRIPTION:Title: Serendipitous decompositions of higher-dimensional continued fractio
ns\nby Anton Lukyanenko (George Mason University) as part of One World
Numeration seminar\n\n\nAbstract\nComplex continued fractions (CFs) repre
sent a complex number using a descending fraction with Gaussian integer co
efficients. The associated dynamical system is exact (Nakada 1981) with a
piecewise-analytic invariant measure (Hensley 2006). Certain higher-dimens
ional CFs\, including CFs over quaternions\, octonions\, as well as the no
n-commutative Heisenberg group can be understood in a unified way using th
e Iwasawa CF framework (L-Vandehey 2022). Under some natural and robust as
sumptions\, ergodicity of the associated systems can then be derived from
a connection to hyperbolic geodesic flow\, but stronger mixing results and
information about the invariant measure remain elusive. Here\, we study I
wasawa CFs under a more delicate serendipity assumption that yields the fi
nite range condition\, allowing us to extend the Nakada-Hensley results to
certain Iwasawa CFs over the quaternions\, octonions\, and in $\\mathbb{R
}^3$.\n \nThis is joint work with Joseph Vandehey.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ronnie Pavlov (University of Denver)
DTSTART:20230425T130000Z
DTEND:20230425T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/114
DESCRIPTION:Title: Subshifts of very low complexity\nby Ronnie Pavlov (University o
f Denver) as part of One World Numeration seminar\n\n\nAbstract\nThe word
complexity function $p(n)$ of a subshift $X$ measures the number of $n$-le
tter words appearing in sequences in $X$\, and $X$ is said to have linear
complexity if $p(n)/n$ is bounded. It's been known since work of Ferenczi
that subshifts X with linear word complexity function (i.e. $\\limsup p(n)
/n$ finite) have highly constrained/structured behavior. I'll discuss rece
nt work with Darren Creutz\, where we show that if $\\limsup p(n)/n < 4/3$
\, then the subshift $X$ must in fact have measurably discrete spectrum\,
i.e. it is isomorphic to a compact abelian group rotation. Our proof uses
a substitutive/S-adic decomposition for such shifts\, and I'll touch on co
nnections to the so-called S-adic Pisot conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Craig S. Kaplan (University of Waterloo)
DTSTART:20230509T120000Z
DTEND:20230509T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/115
DESCRIPTION:Title: An aperiodic monotile\nby Craig S. Kaplan (University of Waterlo
o) as part of One World Numeration seminar\n\n\nAbstract\nA set of shapes
is called aperiodic if the shapes admit tilings of the plane\, but none th
at have translational symmetry. A longstanding open problem asks whether a
set consisting of a single shape could be aperiodic\; such a shape is kno
wn as an aperiodic monotile or sometimes an "einstein". The recently disco
vered "hat" monotile settles this problem in two dimensions. In this talk
I provide necessary background on aperiodicity and related topics in tilin
g theory\, review the history of the search for for an aperiodic monotile\
, and then discuss the hat and its mathematical properties.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Pollicott (University of Warwick)
DTSTART:20230905T120000Z
DTEND:20230905T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/116
DESCRIPTION:Title: Complex Dimensions and Fractal Strings\nby Mark Pollicott (Unive
rsity of Warwick) as part of One World Numeration seminar\n\n\nAbstract\nS
ome years ago M.Lapidus introduced the notion of complex dimensions for a
Cantor set in the real line. These occur as poles of the complex Dirichlet
series formed from the lengths of the bounded intervals (the "fractal str
ings") in the complement of the Cantor set. We will explore further these
ideas when the Cantor set is the attractor of an iterated function scheme
(concentrating on those whose contractions are a finite set of inverse bra
nches of the usual Gauss map).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Worrell (University of Oxford)
DTSTART:20230919T120000Z
DTEND:20230919T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/117
DESCRIPTION:Title: Transcendence of Sturmian Numbers over an Algebraic Base\nby Jam
es Worrell (University of Oxford) as part of One World Numeration seminar\
n\n\nAbstract\nFerenczi and Mauduit showed in 1997 that a number represent
ed over an integer base by a Sturmian sequence of digits is transcendental
. In this talk we generalise this result to hold for all algebraic number
base b of absolute value strictly greater than one. More generally\, for
a given base b and given irrational number θ\, we prove rational linear
independence of the set comprising 1 together with all numbers of the abov
e form whose associated digit sequences have slope θ.\n\nWe give an appli
cation of our main result to the theory of dynamical systems. We show that
for a Cantor set C arising as the set of limit points of a contracted rot
ation f on the unit interval\, where f is assumed to have an algebraic slo
pe\, all elements of C except its endpoints 0 and 1 are transcendental.\n\
nThis is joint work with Florian Luca and Joel Ouaknine.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manfred Madritsch (Université de Lorraine)
DTSTART:20231003T120000Z
DTEND:20231003T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/118
DESCRIPTION:Title: Construction of absolutely normal numbers\nby Manfred Madritsch
(Université de Lorraine) as part of One World Numeration seminar\n\n\nAbs
tract\nLet $b\\geq2$ be a positive integer. Then every real number $x\\in[
0\,1]$ admits a\n$b$-adic representation with digits $a_k$. We call the re
al $x$ simply\nnormal to base $b$ if every digit $d\\in\\{0\,1\,\\dots\,b-
1\\}$ occurs with the same\nfrequency in the $b$-ary representation. Furth
ermore we call $x$ normal to\nbase $b$\, if it is simply normal with respe
ct to $b$\, $b^2$\, $b^3$\, etc.\nFinally we call $x$ absolutely normal if
it is normal with respect to all\nbases $b\\geq2$. \n\nIn the present tal
k we want to generalize this notion to normality in measure\npreserving sy
stems like $\\beta$-expansions and continued fraction expansions.\nThen we
show constructions of numbers that are (absolutely) normal with respect\n
to several different expansions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fumichika Takamizo (Osaka Metropolitan University)
DTSTART:20231017T120000Z
DTEND:20231017T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/119
DESCRIPTION:Title: Finite $\\beta$-expansion of natural numbers\nby Fumichika Takam
izo (Osaka Metropolitan University) as part of One World Numeration semina
r\n\n\nAbstract\nIf $\\beta$ is an integer\, then each $x \\in \\mathbb{Z}
[1/\\beta] \\cap [0\,\\infty)$ has finite expansion in base $\\beta$. As a
generalization of this property for $\\beta>1$\, the condition (F$_{1}$)
that each $x \\in \\mathbb{N}$ has finite $\\beta$-expansion was proposed
by Frougny and Solomyak. \nIn this talk\, we give a sufficient condition f
or (F$_{1}$). Moreover we also find $\\beta$ with property (F$_{1}$) which
does not have positive finiteness property.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Marmi (Scuola Normale Superiore)
DTSTART:20231031T130000Z
DTEND:20231031T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/120
DESCRIPTION:Title: Complexified continued fractions and complex Brjuno and Wilton funct
ions\nby Stefano Marmi (Scuola Normale Superiore) as part of One World
Numeration seminar\n\n\nAbstract\nWe study functions related to the class
ical Brjuno function\, namely k-Brjuno functions and the Wilton function.
Both appear in the study of boundary regularity properties of (quasi) modu
lar forms and their integrals. We then complexify the functional equations
which they fulfill and we construct analytic extensions of the k-Brjuno a
nd Wilton functions to the upper half-plane. We study their boundary behav
iour using an extension of the continued fraction algorithm to the complex
plane. We also prove that the harmonic conjugate of the real k-Brjuno fun
ction is continuous at all irrational numbers and has a decreasing jump of
π/qk at rational points p/q. This is based on joint work with S. B. Lee\
, I. Petrykiewicz and T. I. Schindler\, the paper is available (open sourc
e) at this link: https://link.springer.com/article/10.1007/s00010-023-0096
7-w\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jana Lepšová (Czech Technical University in Prague\, Université
de Bordeaux)
DTSTART:20231114T130000Z
DTEND:20231114T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/121
DESCRIPTION:Title: Dumont-Thomas numeration systems for ℤ\nby Jana Lepšová (Cze
ch Technical University in Prague\, Université de Bordeaux) as part of On
e World Numeration seminar\n\n\nAbstract\nWe extend the well-known Dumont-
Thomas numeration system to ℤ by considering two-sided periodic points o
f a substitution\, thus allowing us to represent any integer in ℤ by a f
inite word (starting with 0 when nonnegative and with 1 when negative). We
show that an automaton returns the letter at position $n \\in ℤ$ of the
periodic point when fed with the representation of $n$. The numeration sy
stem naturally extends to $ℤ^d$. We give an equivalent characterization
of the numeration system in terms of a total order on a regular language.
Lastly\, using particular periodic points\, we recover the well-known two'
s complement numeration system and the Fibonacci analogue of the two's com
plement numeration system.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Bonanno (Università di Pisa)
DTSTART:20231128T130000Z
DTEND:20231128T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/122
DESCRIPTION:Title: Asymptotic behaviour of the sums of the digits for continued fractio
n algorithms\nby Claudio Bonanno (Università di Pisa) as part of One
World Numeration seminar\n\n\nAbstract\nIn this talk I will discuss applic
ations of methods of ergodic theory to obtain pointwise asymptotic behavio
ur for the sum of the digits of some non-regular continued fraction algori
thms. The idea is to study the behaviour of trimmed Birkhoff sums for infi
nite-measure preserving dynamical systems. The talk is based on joint work
with Tanja I. Schindler.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasushi Nagai (Shinshu University)
DTSTART:20231212T130000Z
DTEND:20231212T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/123
DESCRIPTION:Title: Overlap algorithm for general S-adic tilings\nby Yasushi Nagai (
Shinshu University) as part of One World Numeration seminar\n\n\nAbstract\
nWe investigate the question of when a tiling has pure point spectrum\, fo
r the class of $S$-adic tilings\, which includes all self-affine tilings.
The overlap algorithm by Solomyak is a powerful tool to study this problem
for the class of self-affine tilings. We generalize this algorithm for ge
neral $S$-adic tilings\, and apply it to a class of block $S$-adic tilings
to show almost all of them have pure point spectra. This is a joint work
with Jörg Thuswaldner.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karma Dajani (Universiteit Utrecht)
DTSTART:20240116T130000Z
DTEND:20240116T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/124
DESCRIPTION:Title: Alternating N-continued fraction expansions\nby Karma Dajani (Un
iversiteit Utrecht) as part of One World Numeration seminar\n\n\nAbstract\
nWe introduce a family of maps generating continued fractions where the di
git 1 in the numerator is replaced cyclically by some given non-negative i
ntegers $(N_1\, \\dots\, N_m)$. We prove the convergence of the given algo
rithm\, and study the underlying dynamical system generating such expansio
ns. We prove the existence of a unique absolutely continuous invariant erg
odic measure. In special cases\, we are able to build the natural extensio
n and give an explicit expression of the invariant measure. For these case
s\, we formulate a Doeblin-Lenstra type theorem. For other cases we have a
more implicit expression that we conjecture gives the invariant density.
This conjecture is supported by simulations. For the simulations we use a
method that gives us a smooth approximation in every iteration. This is jo
int work with Niels Langeveld.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cathy Swaenepoel (Université Paris Cité)
DTSTART:20240130T130000Z
DTEND:20240130T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/125
DESCRIPTION:Title: Reversible primes\nby Cathy Swaenepoel (Université Paris Cité)
as part of One World Numeration seminar\n\n\nAbstract\nThe properties of
the digits of prime numbers and various other\nsequences of integers have
attracted great interest in recent years.\nFor any positive integer $k$\,
we denote by $\\overleftarrow{k}$ the\nreverse of $k$ in base 2\, defined
by $\\overleftarrow{k} = \\sum_{j=0}^{n-1} \\varepsilon_j\\\,2^{n-1-j}$ wh
ere $k = \\sum_{j=0}^{n-1} \\varepsilon_{j} \\\,2^j$ with $\\varepsilon_j
\\in \\{0\,1\\}$\, $j\\in\\{0\, \\ldots\, n-1\\}$\, $ \\varepsilon_{n-1} =
1$. A natural question is to estimate the number\nof primes $p\\in \\left
[2^{n-1}\,2^n\\right)$ such that\n$\\overleftarrow{p}$ is prime. We will
present a result which provides\nan upper bound of the expected order of m
agnitude. Our method is based\non a sieve argument and also allows us to o
btain a strong lower bound\nfor the number of integers $k$ such that $k$ a
nd $\\overleftarrow{k}$\nhave at most 8 prime factors (counted with multip
licity). We will also\npresent an asymptotic formula for the number of int
egers\n$k\\in \\left[2^{n-1}\,2^n\\right)$ such that $k$ and $\\overleftar
row{k}$\nare squarefree.\n\nThis is a joint work with Cécile Dartyge\, Br
uno Martin\,\nJoël Rivat and Igor Shparlinski.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bartosz Sobolewski (Jagiellonian University in Kraków and Montanu
niversität Leoben)
DTSTART:20240213T130000Z
DTEND:20240213T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/126
DESCRIPTION:Title: Block occurrences in the binary expansion of n and n+t\nby Barto
sz Sobolewski (Jagiellonian University in Kraków and Montanuniversität L
eoben) as part of One World Numeration seminar\n\n\nAbstract\nLet $s(n)$ d
enote the sum of binary digits of a nonnegative integer $n$. In 2012 Cusic
k asked whether for every nonnegative integer $t$ the set of $n$ satisfyin
g $s(n+t) \\geq s(n)$ has natural density strictly greater than $1/2$. So
far it is known that the answer is affirmative for almost all $t$ (in the
sense of density) and $s(n+t) - s(n)$ has approximately Gaussian distribut
ion. During the talk we consider an analogue of this problem concerning th
e function $r(n)$\, which counts the occurrences of the block $11$ in the
binary expansion of $n$. In particular\, we prove that the distribution o
f $r(n+t)-r(n)$ is approximately Gaussian as well. We also discuss a gener
alization to an arbitrary block of binary digits. This is a joint work wit
h Lukas Spiegelhofer.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joël Ouaknine (Max Planck Institute for Software Systems)
DTSTART:20240312T130000Z
DTEND:20240312T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/127
DESCRIPTION:Title: The Skolem Landscape\nby Joël Ouaknine (Max Planck Institute fo
r Software Systems) as part of One World Numeration seminar\n\n\nAbstract\
nThe Skolem Problem asks how to determine algorithmically whether a given
linear recurrence sequence (such as the Fibonacci numbers) has a zero. It
is a central question in dynamical systems and number theory\, and has man
y connections to other branches of mathematics and computer science. Unfor
tunately\, its decidability has been open for nearly a century! In this ta
lk\, I will present a survey of what is known on the Skolem Problem and re
lated questions\, including recent and ongoing developments.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikita Shulga (La Trobe University)
DTSTART:20240326T130000Z
DTEND:20240326T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/128
DESCRIPTION:Title: Radical bound for Zaremba’s conjecture\nby Nikita Shulga (La T
robe University) as part of One World Numeration seminar\n\n\nAbstract\nZa
remba's conjecture states that for each positive integer $q$\, there exist
s a coprime integer $a$\, smaller than $q$\, such that partial quotients i
n the continued fraction expansion of $a/q$ are bounded by some absolute c
onstant. Despite major breakthroughs in the recent years\, the conjecture
is still open. In this talk I will discuss a new result towards Zaremba's
conjecture\, proving that for each denominator\, one can find a numerator\
, such that partial quotients are bounded by the radical of the denominato
r\, i.e. the product of distinct prime factors. This generalizes the resul
t by Niederreiter and improves upon some results of Moshchevitin-Murphy-Sh
kredov.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Usuki (Kyoto University)
DTSTART:20240423T130000Z
DTEND:20240423T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/130
DESCRIPTION:Title: On a lower bound of the number of integers in Littlewood's conjectur
e\nby Shunsuke Usuki (Kyoto University) as part of One World Numeratio
n seminar\n\n\nAbstract\nLittlewood's conjecture is a famous and long-stan
ding open problem which states that\, for every $(\\alpha\,\\beta) \\in \\
mathbb{R}^2$\, $n\\|n\\alpha\\|\\|n\\beta\\|$ can be arbitrarily small for
some integer $n$.\nThis problem is closely related to the action of diago
nal matrices on $\\mathrm{SL}(3\,\\mathbb{R})/\\mathrm{SL}(3\,\\mathbb{Z})
$\, and a groundbreaking result was shown by Einsiedler\, Katok and Linden
strauss from the measure rigidity for this action\, saying that Littlewood
's conjecture is true except on a set of Hausdorff dimension zero.\nIn thi
s talk\, I will explain about a new quantitative result on Littlewood's co
njecture which gives\, for every $(\\alpha\,\\beta) \\in \\mathbb{R}^2$ ex
cept on sets of small Hausdorff dimension\, an estimate of the number of i
ntegers $n$ which make $n\\|n\\alpha\\|\\|n\\beta\\|$ small. The keys for
the proof are the measure rigidity and further studies on behavior of empi
rical measures for the diagonal action.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Kempton (University of Manchester)
DTSTART:20240507T120000Z
DTEND:20240507T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/131
DESCRIPTION:Title: The Dynamics of the Fibonacci Partition Function\nby Tom Kempton
(University of Manchester) as part of One World Numeration seminar\n\n\nA
bstract\nThe Fibonacci partition function $R(n)$ counts the number of ways
of representing a natural number $n$ as the sum of distinct Fibonacci num
bers. For example\, $R(6)=2$ since $6=5+1$ and $6=3+2+1$. An explicit form
ula for $R(n)$ was recently given by Chow and Slattery. In this talk we ex
press $R(n)$ in terms of ergodic sums over an irrational rotation\, which
allows us to prove lots of statements about the local structure of $R(n)$.
\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaétan Guillot (Université Paris-Saclay)
DTSTART:20240521T120000Z
DTEND:20240521T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/132
DESCRIPTION:Title: Approximation of linear subspaces by rational linear subspaces\n
by Gaétan Guillot (Université Paris-Saclay) as part of One World Numerat
ion seminar\n\n\nAbstract\nWe elaborate on a problem raised by Schmidt in
1967: rational approximation of linear subspaces of $\\mathbb{R}^n$. In or
der to study the quality approximation of irrational numbers by rational o
nes\, one can introduce the exponent of irrationality of a number. We can
then generalize this notion in the framework of vector subspaces for the a
pproximation of a subspace by so-called rational subspaces.\n\nAfter brief
ly introducing the tools for constructing this generalization\, I will pre
sent the different possible studies of this object. Finally I will explain
how we can construct spaces with prescribed exponents.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noy Soffer Aranov (Technion)
DTSTART:20240618T120000Z
DTEND:20240618T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/133
DESCRIPTION:Title: Escape of Mass of the Thue Morse Sequence\nby Noy Soffer Aranov
(Technion) as part of One World Numeration seminar\n\n\nAbstract\nOne way
to study the distribution of quadratic number fields is through the evolut
ion of continued fraction expansions. In the function field setting\, it w
as shown by de Mathan and Teullie that given a quadratic irrational $\\The
ta$\, the degrees of the periodic part of the continued fraction of $t^n\\
Theta$ are unbounded. Paulin and Shapira improved this by proving that qua
dratic irrationals exhibit partial escape of mass. Moreover\, they conject
ured that they must exhibit full escape of mass. We show that the Thue Mor
se sequence is a counterexample to their conjecture. In this talk we shall
discuss the technique of proof as well as the connection between escape o
f mass in continued fractions\, Hecke trees\, and number walls. This is pa
rt of ongoing work joint with Erez Nesharim.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Kristensen (Aarhus Universitet)
DTSTART:20240917T120000Z
DTEND:20240917T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/134
DESCRIPTION:Title: On the distribution of sequences of the form $(q_n y)$\nby Simon
Kristensen (Aarhus Universitet) as part of One World Numeration seminar\n
\n\nAbstract\nThe distribution of sequences of the form $(q_n y)$ with $(q
_n)$ a sequence of integers and $y$ a real number have attracted quite a b
it of attention\, for instance due to their relation to inhomogeneous Litt
lewood type problems. In this talk\, we will provide some results on the L
ebesgue measure and Hausdorff dimension on the set of points in the unit i
nterval approximated to a certain rate by points from such a sequence. A f
eature of our approach is that we obtain estimates even in the case when t
he sequence $(q_n)$ grows rather slowly. This is joint work with Tomas Per
sson.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dong Han Kim (Dongguk University)
DTSTART:20241001T120000Z
DTEND:20241001T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/135
DESCRIPTION:Title: Uniform Diophantine approximation on the Hecke group $H_4$\nby D
ong Han Kim (Dongguk University) as part of One World Numeration seminar\n
\n\nAbstract\nDirichlet's uniform approximation theorem is a fundamental r
esult in Diophantine approximation that gives an optimal rate of approxima
tion.\nWe study uniform Diophantine approximation properties on the Hecke
group $H_4$ in terms of the Rosen continued fractions.\nFor a given real n
umber $\\alpha$\, the best approximations are convergents of the Rosen con
tinued fraction and the dual Rosen continued fraction of $\\alpha$.\nWe gi
ve analogous theorems of Dirichlet uniform approximation and the Legendre
theorem with optimal constants.\nThis is joint work with Ayreena Bakhtawar
and Seul Bee Lee.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (Stellenbosch University)
DTSTART:20241112T130000Z
DTEND:20241112T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/136
DESCRIPTION:Title: On a question of Douglass and Ono\nby Florian Luca (Stellenbosch
University) as part of One World Numeration seminar\n\n\nAbstract\nIt is
known that the partition function $p(n)$ obeys Benford's law in any integ
er base $b\\ge 2$. A similar result was obtained by Douglass and Ono for t
he plane partition function $\\text{PL}(n)$ in a recent paper. In their pa
per\, Douglass and Ono asked for an explicit version of this result. In pa
rticular\, given an integer base $b\\ge 2$ and string $f$ of digits in bas
e $b$ they asked for an explicit value $N(b\,f)$ such that there exists $n
\\le N(b\,f)$ with the property that $\\text{PL}(n)$ starts with the strin
g $f$ when written in base $b$. In my talk\, I will present an explicit va
lue for $N(b\,f)$ both for the partition function $p(n)$ as well as for th
e plane partition function $\\text{PL}(n)$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haojie Ren (Technion)
DTSTART:20241126T130000Z
DTEND:20241126T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/137
DESCRIPTION:Title: The dimension of Bernoulli convolutions in $\\mathbb{R}^d$\nby H
aojie Ren (Technion) as part of One World Numeration seminar\n\n\nAbstract
\nFor $(\\lambda_{1}\,\\dots\,\\lambda_{d})=\\lambda\\in(0\,1)^{d}$ with $
\\lambda_{1}>\\cdots>\\lambda_{d}$\,\ndenote by $\\mu_{\\lambda}$ the Bern
oulli convolution associated to\n$\\lambda$. That is\, $\\mu_{\\lambda}$ i
s the distribution of the random\nvector $\\sum_{n\\ge0}\\pm\\left(\\lambd
a_{1}^{n}\,\\dots\,\\lambda_{d}^{n}\\right)$\,\nwhere the $\\pm$ signs are
chosen independently and with equal weight.\nAssuming for each $1\\le j\\
le d$ that $\\lambda_{j}$ is not a root\nof a polynomial with coefficients
$\\pm1\,0$\, we prove that the dimension\nof $\\mu_{\\lambda}$ equals $\\
min\\{ \\dim_{L}\\mu_{\\lambda}\,d\\} $\,\nwhere $\\dim_{L}\\mu_{\\lambda}
$ is the Lyapunov dimension. This is a joint work with Ariel Rapaport.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Robertson (University of Manchester)
DTSTART:20241015T120000Z
DTEND:20241015T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/138
DESCRIPTION:Title: Low Discrepancy Digital Hybrid Sequences and the $t$-adic Littlewood
Conjecture\nby Steven Robertson (University of Manchester) as part of
One World Numeration seminar\n\n\nAbstract\nThe discrepancy of a sequence
measures how quickly it approaches a uniform distribution. Given a natura
l number $d$\, any collection of one-dimensional so-called low discrepancy
sequences $\\{S_i : 1 \\le i \\le d\\}$ can be concatenated to create a $
d$-dimensional hybrid sequence $(S_1\, . . . \, S_d)$. Since their introdu
ction by Spanier in 1995\, many connections between the discrepancy of a h
ybrid sequence and the discrepancy of its component sequences have been di
scovered. However\, a proof that a hybrid sequence is capable of being low
discrepancy has remained elusive. In this talk\, an explicit connection b
etween Diophantine approximation over function fields and two dimensional
low discrepancy hybrid sequences is provided. \n\nSpecifically\, it is sho
wn that any counterexample to the so-called $t$-adic Littlewood Conjecture
($t$-LC) can be used to create a low discrepancy digital Kronecker-Van de
r Corput sequence. Such counterexamples to $t$-LC are known explicitly ov
er a number of finite fields by\, on the one hand\, Adiceam\, Nesharim and
Lunnon\, and on the other\, by Garrett and the Robertson. All necessary c
oncepts will be defined in the talk.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Shirandami (University of Manchester)
DTSTART:20241029T130000Z
DTEND:20241029T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/139
DESCRIPTION:Title: Probabilistic Effectivity in the Subspace Theorem and the Distributi
on of Algebraic Projective Points\nby Victor Shirandami (University of
Manchester) as part of One World Numeration seminar\n\n\nAbstract\nThe ce
lebrated Roth’s theorem in Diophantine Approximation determines the degr
ee to which an algebraic number may be approximated by rationals. A coroll
ary of this theorem yields a transcendence criterion for real numbers base
d off of their decimal expansion. This theorem\, and its broad generalisat
ion due to Schmidt\, famously suffers from ineffectivity. This motivates o
ne to address this issue in the probabilistic context\, whereby one makes
progress in the direction of effectivity in an appropriately defined proba
bilistic regime. From this analysis is derived an analogue of Khintchine's
theorem for algebraic numbers\, answering a question of Beresnevich\, Ber
nick\, and Dodson on a density version of Waldschmidt’s conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Messaoudi (Universidade Estadual Paulista)
DTSTART:20241210T123000Z
DTEND:20241210T131500Z
DTSTAMP:20260404T131146Z
UID:OWNS/140
DESCRIPTION:Title: Adding machine\, automata and Julia sets\nby Ali Messaoudi (Univ
ersidade Estadual Paulista) as part of One World Numeration seminar\n\n\nA
bstract\nA stochastic adding machine (defined by P.R. Killeen and T.J. Tay
lor) is a Markov chain whose states are natural integers\, which models th
e process of adding the number 1 but where there is a probability of failu
re in which a carry is not performed when necessary. In this lecture\, we
will talk about dynamical\, spectral and probabilistic properties of exten
sions for the stochastic adding machine and their connections with other t
opics as Julia sets\, Automata and Dynamical Systems on Banach spaces.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Paul Allouche (CNRS\, Sorbonne Université)
DTSTART:20250107T130000Z
DTEND:20250107T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/141
DESCRIPTION:Title: Kolam\, Ethnomathematics\, and Morphisms\nby Jean-Paul Allouche
(CNRS\, Sorbonne Université) as part of One World Numeration seminar\n\n\
nAbstract\nKolam is a form of traditional decorative art in India\, that i
s drawn by using rice flour\, white stone powder\, chalk or chalk powder.
It is often practised by women in front of their house entrance. One parti
cular kolam uses a 4x4 grid. It can also be found in the Vanuatu Islands\,
in Africa\, etc. We show that a natural generalization on grids of size 8
x8\, 16x16\, etc. is linked to... the Thue-Morse sequence. Further we unve
il two (twin) morphisms that generate this family of kolam\, and show that
they appear in unrelated and somewhat unexpected fields. Time permitting
we will allude to a vast\, relatively new\, field: ethnomathematic(s).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Garrity (Williams College)
DTSTART:20250121T130000Z
DTEND:20250121T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/142
DESCRIPTION:Title: Multi-dimensional continued fractions and integer partitions: Using
the Natural Extension to create a tree structure on partitions\nby Tho
mas Garrity (Williams College) as part of One World Numeration seminar\n\n
Abstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giulia Salvatori (Politecnico di Torino)
DTSTART:20250204T130000Z
DTEND:20250204T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/143
DESCRIPTION:Title: Continued Fractions\, Quadratic Forms\, and Regulator Computation fo
r Integer Factorization\nby Giulia Salvatori (Politecnico di Torino) a
s part of One World Numeration seminar\n\n\nAbstract\nIn the realm of inte
ger factorization\, certain methods\, such as CFRAC\, leverage the propert
ies of continued fractions\, while others\, like SQUFOF\, combine these pr
operties with the tools provided by quadratic forms. Recently\, Michele El
ia revisited the fundamental concepts of SQUFOF\, including reduced quadra
tic forms\, distance between quadratic forms\, and Gauss composition\, off
ering a new perspective for designing factorization methods.\n\nIn this se
minar\, we present our algorithm\, which is a refinement of Elia's method\
, along with a precise analysis of its computational cost.\nOur algorithm
is polynomial-time\, provided knowledge of a (not too large) multiple of t
he regulator of $\\mathbb{Q}(\\sqrt{N})$.\nThe computation of the regulato
r governs the total computational cost\, which is subexponential\, and in
particular $O(\\exp(\\frac{3}{\\sqrt{8}}\\sqrt{\\ln N \\ln \\ln N}))$. \nT
his makes our method more efficient than CFRAC and SQUFOF\, though less ef
ficient than the General Number Field Sieve.\n\nWe identify a broad family
of integers to which our method is applicable including certain classes o
f RSA moduli.\nFinally\, we introduce some promising avenues for refining
our method. These span several areas\, ranging from Algebraic Number Theor
y\, particularly for estimating the size of the regulator of $\\mathbb{Q}(
\\sqrt{N})$\, to Analytic Number Theory\, particularly for computing a spe
cific class of $L$-functions.\n\nJoint work with Nadir Murru.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neil MacVicar (Queen's University)
DTSTART:20250218T130000Z
DTEND:20250218T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/144
DESCRIPTION:Title: Intersecting Cantor sets generated by Complex Radix Expansions\n
by Neil MacVicar (Queen's University) as part of One World Numeration semi
nar\n\n\nAbstract\nConsider the classical middle third Cantor set. This is
a self-similar set containing all the numbers in the unit interval which
have a ternary expansion that avoids the digit 1. We can ask when the inte
rsection of the Cantor set with a translate of itself is also self-similar
. Sufficient and necessary conditions were given by Deng\, He\, and Wen in
2008. This question has also been generalized to classes of subsets of th
e unit interval. I plan to discuss how existing ideas can be used to addre
ss the question for certain self-similar sets with dimension greater than
one. These ideas will be illustrated using a class of self-similar sets in
the plane that can be realized as radix expansions in base $-n+i$ where $
n$ is a positive integer. I will also discuss a property of the fractal di
mensions of these kinds of intersections.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Ovsienko (CNRS\, Université de Reims-Champagne-Ardenne)
DTSTART:20250318T130000Z
DTEND:20250318T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/145
DESCRIPTION:Title: From Catalan numbers to integrable dynamics: continued fractions and
Hankel determinants for q-numbers\nby Valentin Ovsienko (CNRS\, Unive
rsité de Reims-Champagne-Ardenne) as part of One World Numeration seminar
\n\n\nAbstract\nThe classical Catalan and Motzkin numbers have remarkable
continued fraction expansions\, the corresponding sequences of Hankel dete
rminants consist of -1\, 0 and 1 only. We find an infinite family of power
series corresponding to q-deformed real numbers that have very similar pr
operties. Moreover\, their sequences of Hankel determinants turn out to sa
tisfy Somos and Gale-Robinson recurrences. (Partially based on a joint wor
k with Emmanuel Pedon.)\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meng Wu (Oulun yliopisto)
DTSTART:20250401T120000Z
DTEND:20250401T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/146
DESCRIPTION:Title: On normal numbers in fractals\nby Meng Wu (Oulun yliopisto) as p
art of One World Numeration seminar\n\n\nAbstract\nLet $K$ be the ternary
Cantor set\, and let $\\mu$ be the Cantor–Lebesgue measure on $K$. It is
well known that every point in $K$ is not 3-normal. However\, if we take
any natural number $p \\ge 2$ that is not a power of 3\, then $\\mu$-almos
t every point in $K$ is $p$-normal. This classical result is due to Cassel
s and W. Schmidt.\n\nAnother way to obtain normal numbers from K is by res
caling and translating $K$\, then examining the transformed set. A recent
nice result by Dayan\, Ganguly\, and Barak Weiss shows that for any irrati
onal number $t$\, for $\\mu$-almost all $x \\in K$\, the product $tx$ is 3
-normal.\n\nIn this talk\, we will discuss these results and their general
izations\, including replacing $p$ with an arbitrary beta number and consi
dering more general times-3 invariant measures instead of the Cantor–Leb
esgue measure.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Cumberbatch (Purdue University)
DTSTART:20250415T120000Z
DTEND:20250415T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/147
DESCRIPTION:Title: Smooth numbers with restricted digits\nby James Cumberbatch (Pur
due University) as part of One World Numeration seminar\n\n\nAbstract\nInt
egers obeying a digital restriction\, such as having no 7s in their base 1
0 representation\, are a discrete analog of the Cantor set and have been a
recent topic of interest in analytic number theory. Smooth integers\, whi
ch are integers having only small prime factors\, are an important class o
f integers with applications to many different areas of math. In this talk
\, we find an asymptotic on the intersection between the two.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yan Huang (Chongqing University)
DTSTART:20250429T120000Z
DTEND:20250429T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/148
DESCRIPTION:Title: The Coincidence of Rényi–Parry Measures for β-Transformation
\nby Yan Huang (Chongqing University) as part of One World Numeration semi
nar\n\n\nAbstract\nWe present a complete characterization of two non-integ
ers with the same Rényi-Parry measure.\nWe prove that for two non-integer
s $\\beta_1 \,\\beta_2 >1$\, the Rényi-Parry measures coincide if and on
ly if $\\beta_1$ is the root of equation $x^2-qx-p=0$\, where $p\,q\\in\\m
athbb{N}$ with $p\\leq q$\, and $\\beta_2 = \\beta_1 + 1$\, which confirms
a conjecture of Bertrand-Mathis in [A. Bertrand-Mathis\, Acta Math. Hunga
r. 78\, no. 1-2 (1998):71–78].\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artem Dudko (IM PAN)
DTSTART:20250513T120000Z
DTEND:20250513T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/149
DESCRIPTION:Title: On attractors of Fibonacci maps\nby Artem Dudko (IM PAN) as part
of One World Numeration seminar\n\n\nAbstract\nIn 1990s Bruin\, Keller\,
Nowicki\, and van Strien showed that smooth unimodal maps with Fibonacci c
ombinatorics and sufficiently high degree of a critical point have a wild
attractor\, i.e. their metric and topological attractors do not coincide.
However\, until now there were no reasonable estimates on the degree of th
e critical point needed.\n\nIn the talk I will present an approach for stu
dying attractors of maps\, which are periodic points of a renormalization.
Using this approach and rigorous computer estimates\, we show that the Fi
bonacci map of degree $d=3.8$ does not have a wild attractor\, but that fo
r degree $d=5.1$ the wild attractor exists. The talk is based on a joint w
ork with Denis Gaidashev.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Savinien Kreczman (Université de Liège)
DTSTART:20250527T120000Z
DTEND:20250527T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/150
DESCRIPTION:Title: Positionality for Dumont-Thomas numeration systems\nby Savinien
Kreczman (Université de Liège) as part of One World Numeration seminar\n
\n\nAbstract\nDumont-Thomas numeration systems are a subclass of abstract
numeration systems where the factorisation of the fixed point of a substit
ution is used to represent numbers. A positional numeration system is one
where a weight can be assigned to each position so that the evaluation map
is an inner product with the weights. For general abstract numeration sys
tems\, deciding positionality is an open problem. In this talk\, we define
an extension of Dumont-Thomas numeration systems to all integers. We then
offer a criterion for deciding the positionality of such a system. If tim
e permits\, we show a link to Bertrand numeration systems\, another famili
ar class of numeration systems.\n\nJoint work with Sébastien Labbé and M
anon Stipulanti.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuta Suzuki (Rikkyo University)
DTSTART:20250610T120000Z
DTEND:20250610T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/151
DESCRIPTION:Title: Telhcirid's theorem on arithmetic progressions\nby Yuta Suzuki (
Rikkyo University) as part of One World Numeration seminar\n\n\nAbstract\n
The classical Dirichlet theorem on arithmetic progressions states that the
re are infinitely many primes in a given arithmetic progression with a tri
vial necessary condition. In this talk\, we prove a "reversed" version of
this theorem\, which may be called Telhcirid's theorem on arithmetic progr
essions\, i.e.\, we prove that there are infinitely many primes whose reve
rse of radix representation is in a given arithmetic progression except in
some degenerate cases. This is a joint work with Gautami Bhowmik (Univers
ity of Lille).\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vilmos Komornik (Université de Strasbourg)
DTSTART:20250923T120000Z
DTEND:20250923T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/152
DESCRIPTION:Title: Topology of univoque sets in double-base expansions\nby Vilmos K
omornik (Université de Strasbourg) as part of One World Numeration semina
r\n\n\nAbstract\nThis is a joint work with Yuru Zou and YiChang Li. Given
two real numbers $q_0\,q_1>1$ satisfying $q_0+q_1\\geq q_0q_1$ and two rea
l numbers $d_0\\ne d_1$\, by a double-base expansion of a real number $x$
we mean a sequence $(i_k)\\in \\{0\,1\\}^{\\infty}$ such that\n\\[\nx=\\su
m_{k=1}^{\\infty}\\frac{d_{i_k}}{q_{{i_1}}q_{{i_2}}\\cdots q_{{i_k}}}.\n\\
]\nWe denote by $\\mathcal{U}_{{q_0\,q_1}}$ the set of numbers $x$ having
a unique expansion.\nThe topological properties of $\\mathcal{U}_{{q_0\,
q_1}}$ have been investigated in the equal-base case $q_0=q_1$ for a long
time. \nWe extend this research to the case $q_0\\neq q_1$. \nWhile many
results remain valid\, a great number of new phenomena appear due to the
increased complexity of double-base expansions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Chiara Lai (Sapienza Università di Roma)
DTSTART:20251007T110000Z
DTEND:20251007T120000Z
DTSTAMP:20260404T131146Z
UID:OWNS/153
DESCRIPTION:Title: Optimal expansions of Kakeya sequences\nby Anna Chiara Lai (Sapi
enza Università di Roma) as part of One World Numeration seminar\n\n\nAbs
tract\nExpansions of Kakeya sequences generalize the expansions in non-int
eger bases and they display analogous redundancy phenomena. In this talk\,
we present a characterization of optimal expansions of Kakeya sequences\,
and we provide conditions for the existence of unique expansions with res
pect to Kakeya sequences.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bo Wang (Sun Yat-Sen University\, Jiaying University)
DTSTART:20251021T120000Z
DTEND:20251021T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/154
DESCRIPTION:Title: Ramírez’s Problems and Fibers on Well Approximable Set of Systems
of Affine Forms\nby Bo Wang (Sun Yat-Sen University\, Jiaying Univers
ity) as part of One World Numeration seminar\n\n\nAbstract\nIn this talk\,
we show that badly approximable matrices are exactly those that\, for eve
ry inhomogeneous parameter\, cannot be inhomogeneous approximated at every
monotone divergent rate\, which generalizes Ramírez's result (2018). We
also establish some metrical results of the fibers on well approximable se
t of systems of affine forms\, which gives answers to three of Ramírez's
problems (2018). Furthermore\, we prove that badly approximable systems ar
e exactly those that for each monotone convergent rate $\\psi$ cannot be a
pproximated at $\\psi$. Moreover\, we study the topological structure of t
he set of approximation functions. This is a joint work with Bing Li.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Colin Faverjon (CNRS\, Université de Picardie Jules Verne)
DTSTART:20251104T130000Z
DTEND:20251104T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/155
DESCRIPTION:Title: Writing numbers in multiple bases: the viewpoint of finite automata<
/a>\nby Colin Faverjon (CNRS\, Université de Picardie Jules Verne) as par
t of One World Numeration seminar\n\n\nAbstract\nAlthough binary and decim
al representations of numbers coexist seamlessly in our digital world\, th
ese changes of basis conceal profound mysteries. Consider\, for example\,
the following statements\, both currently out of reach:
\n- The date w
hen Trump will leave the US presidency appears in the decimal expansion of
every sufficiently large power of 2\;
\n- The real number whose binar
y expansion is the characteristic sequence of powers of 3 contains the pat
tern 1312 infinitely often in its decimal expansion.\n
\nBoth statement
s rely on the heuristic that expansions in multiplicatively independent ba
ses (such as 2 and 10) should share no common structure. Furstenberg captu
red this heuristic through a series of results and conjectures concerning
the joint behavior of the dynamical systems ×p and ×q on the torus.\n\nI
n this talk\, we approach this question from the perspective pioneered by
Turing\, Hartmanis\, Stearns\, and Cobham: that of computational complexit
y. Powers of 2 are particularly easy to recognize from their base-2 expans
ion—a task achievable by a finite automaton. Cobham's theorem then impli
es that no automaton can recognize them from their decimal expansion. Simi
larly\, one can readily construct a finite automaton with output that prod
uces the binary expansion of the real number introduced above. Whether the
re exists another automaton producing its decimal expansion remained open
until recently.\n\nIn this talk\, we present how this question has been so
lved using a transcendence method known as Mahler's method. While this app
roach yields a new proof and an algebraic generalization of Cobham's theor
em\, its main contribution is the following statement: no irrational re
al number has expansions in two multiplicatively independent bases that ca
n both be produced by finite automata.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Glendinning (University of Manchester)
DTSTART:20251118T130000Z
DTEND:20251118T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/156
DESCRIPTION:Title: Positive Hausdorff dimension for the survivor set and transitions to
chaos for piecewise smooth maps\nby Paul Glendinning (University of M
anchester) as part of One World Numeration seminar\n\n\nAbstract\nWe consi
der two related problems. The transition to chaos in the sense of positive
topological entropy for one-dimensional piecewise smooth maps\, and the t
ransition to positive Hausdorff dimension for the survivor set of associat
ed open maps. We describe an iterative process that determines the boundar
ies of positive topological entropy (resp. positive Hausdorff dimension).
The boundary can then be characterised via substitution sequences that gen
eralise the Thue-Morse sequence for continuous maps of the interval. This
work is joint with Clément Hege.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michał Rams (IM PAN)
DTSTART:20251216T130000Z
DTEND:20251216T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/157
DESCRIPTION:Title: The entropy of Lyapunov-optimizing measures for some matrix cocycles
\nby Michał Rams (IM PAN) as part of One World Numeration seminar\n\n
\nAbstract\nConsider a simple (to formulate...) mathematical object: you a
re given a finite collection of matrices $A_i\\in GL(2\,\\mathbb R)\; i=1\
,\\ldots\,k$ and you are allowed to multiply them\, in any order. The noti
on you are interested in is the exponential rate of speed of growth of the
norm: given $\\omega\\in \\{1\,\\ldots\,k\\}^{\\mathbb N}$\, let\n\\[\n\\
lambda(\\omega) = \\lim_{n\\to\\infty} \\frac 1n \\log ||A_{\\omega_n} \\c
dot \\ldots \\cdot A_{\\omega_1}||.\n\\] \nThis object has many names\, in
dynamical systems we call it the Lyapunov exponent. \n\nWe are in particu
lar interested in the set of those $\\omega$'s that give the extremal (max
imal\, minimal) value of the Lyapunov exponent. A long-standing conjecture
states that for a generic matrix collection those sets ought to be {\\it
small}\, in some sense. In the result I will present we (Jairo Bochi and m
e) are proving that for certain open set of collections of matrices those
$\\omega$'s that maximize/minimize Lyapunov exponent have zero topological
entropy.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shu-Qin Zhang (Zhengzhou University)
DTSTART:20251202T130000Z
DTEND:20251202T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/158
DESCRIPTION:Title: When the conformal dimension of a self-affine sponge of Lalley type
is zero\nby Shu-Qin Zhang (Zhengzhou University) as part of One World
Numeration seminar\n\n\nAbstract\nA compact metric space $X$ is uniformly
disconnected if there exists $\\delta_0$ such that there is no $\\delta_0$
-sequence\, which is a sequence of points $(x_0\,x_1\,\\dots\,x_n)$ satisf
ying $\\rho(x_{i-1}\,x_{i})\\leq \\delta_0 \\rho(x_0\,x_n)$ for all $1\\l
eq i\\leq n$. We present two main results. First\, we give a necessary a
nd sufficient condition for a diagonal self-affine sponge of Lalley-Gatzou
ras type to be uniformly disconnected. Second\, we show that $K$ is unifor
mly disconnected if and only if the conformal dimension of $K$ is $0$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joaco Prandi (University of Waterloo)
DTSTART:20260113T130000Z
DTEND:20260113T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/159
DESCRIPTION:Title: When the weak separation condition implies the generalize finite typ
e in $\\mathbb{R}^d$\nby Joaco Prandi (University of Waterloo) as part
of One World Numeration seminar\n\n\nAbstract\nLet $S$ be an iterated fun
ction system with full support. Under some restrictions on the allowable r
otations\, we will show that $S$ satisfies the weak separation condition i
f and only if it satisfies the generalized finite-type condition. To do th
is\, we will extend the notion of net intervals from $\\mathbb{R}$ to $\\m
athbb{R}^d$. If time allows\, we will also use net intervals to calculate
the local dimension of a self-similar measure with the finite-type conditi
on and full support. This talk is based on joint work with Kevin G. Hare.\
n
LOCATION:https://stable.researchseminars.org/talk/OWNS/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henri Cohen (Université de Bordeaux)
DTSTART:20260127T130000Z
DTEND:20260127T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/160
DESCRIPTION:Title: Continued Fractions and Irrationality Measures for Chowla-Selberg Ga
mma Quotients\nby Henri Cohen (Université de Bordeaux) as part of One
World Numeration seminar\n\n\nAbstract\nWe give 39 rapidly convergent con
tinued fractions for Chowla--Selberg gamma quotients\, and deduce good irr
ationality measures for 20 of them\, including for $\\operatorname{CS}(-3)
=(\\Gamma(1/3)/\\Gamma(2/3))^3$\, for $a^{1/4}\\operatorname{CS}(-4)=a^{1/
4}(\\Gamma(1/4)/\\Gamma(3/4))^2$ with $a=12$ and $a=1/5$\, and for $\\oper
atorname{CS}(-7)=\\Gamma(1/7)\\Gamma(2/7)\\Gamma(4/7)/(\\Gamma(3/7)\\Gamma
(5/7)\\Gamma(6/7))$.\nThese appear to be the first proved and reasonable i
rrationality measures for\ngamma quotients.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junnosuke Koizumi (RIKEN iTHEMS)
DTSTART:20260210T130000Z
DTEND:20260210T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/161
DESCRIPTION:Title: Irrationality Sequences\nby Junnosuke Koizumi (RIKEN iTHEMS) as
part of One World Numeration seminar\n\n\nAbstract\nSometimes one can prov
e the irrationality of the sum of reciprocals of a sequence of positive in
tegers using only information about the growth rate of the sequence. Erdő
s and Straus introduced the notion of an irrationality sequence in order t
o isolate nontrivial aspects of this relationship. Despite its elementary
formulation\, the theory of irrationality sequences still contains many op
en problems. For instance\, the question of whether $2^{2^n}$ is a (type 2
) irrationality sequence is a particularly interesting open problem. Recen
tly\, Kovač and Tao obtained several interesting results on the asymptoti
c behavior of irrationality sequences. We study sums of reciprocals of dou
bly exponential sequences and show\, among other results\, that there are
at most countably many real numbers $a>1$ for which $a^{2^n}$ is a (type 2
) irrationality sequence. We also explain how such questions are related t
o certain greedy Egyptian fraction expansions.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pascal Jelinek (Montanuniversität Leoben)
DTSTART:20260224T130000Z
DTEND:20260224T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/162
DESCRIPTION:Title: Ratio of sum of digits functions in two bases\nby Pascal Jelinek
(Montanuniversität Leoben) as part of One World Numeration seminar\n\n\n
Abstract\nIn 2019 La Bretèche\, Stoll and Tenenbaum showed that the ratio
of the sum of digits function $s_p(n)/s_q(n)$ of two multiplicatively ind
ependent bases $p$ and $q$ is dense in $\\mathbb{Q}^+$. Spiegelhofer prove
d that when $p = 2$ and $q = 3$\, the ratio 1 is attained infinitely many
times\, which he extended jointly with Drmota to arbitrary values in $\\ma
thbb{Q}^+$. In this talk\, I generalize this result further\, showing that
for two arbitrary multiplicatively independent bases\, $s_p(n)/s_q(n)$ at
tains every value in $\\mathbb{Q}^+$ infinitely many times.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shintaro Suzuki (Tokyo Gakugei University)
DTSTART:20260310T130000Z
DTEND:20260310T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/163
DESCRIPTION:Title: Hausdorff dimension of digit frequency sets for beta-expansions and
a generalization of the Hata-Yamaguchi formula\nby Shintaro Suzuki (To
kyo Gakugei University) as part of One World Numeration seminar\n\n\nAbstr
act\nWe consider the digit frequency set of the digit 1 for beta-expansion
s in the case of $1<\\beta\\leq2$ and give an exact formula for its Hausdo
rff dimension via transfer operator method. As a related topic\,\nwe intro
duce a generalization of the Lebesgue singular function and show that \nit
satisfies a version of the Hata-Yamaguchi formula\, which yields a Takagi
-like function for beta-expansions with the base $1<\\beta<2$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuru Zou (Shenzhen University)
DTSTART:20260324T130000Z
DTEND:20260324T140000Z
DTSTAMP:20260404T131146Z
UID:OWNS/164
DESCRIPTION:Title: Hausdorff dimension of double base expansions and binary shifts with
a hole\nby Yuru Zou (Shenzhen University) as part of One World Numera
tion seminar\n\n\nAbstract\nFor two real bases $q_0\, q_1 > 1$\, a binary
sequence $i_1 i_2 \\cdots \\in \\{0\,1\\}^\\infty$ is called the $(q_0\,q_
1)$-expansion of the number\n\n\\[\n\\pi_{q_0\,q_1}(i_1 i_2 \\cdots) = \\s
um_{k=1}^\\infty \\frac{i_k}{q_{i_1} \\cdots q_{i_k}}.\n\\]\nLet $\\mathca
l{U}_{q_0\,q_1}$ denote the set of all real numbers having a unique $(q_0\
,q_1)$-expansion. When the two bases coincide\, i.e.\, $q_0 = q_1 = q$\, i
t was shown by Allaart and Kong (2019) that the Hausdorff dimension of the
univoque set $\\mathcal{U}_{q\,q}$ varies continuously in $q$\, building
on earlier work of Komornik\, Kong\, and Li (2017). In this talk\, we will
derive explicit formulas for the Hausdorff dimension of $\\mathcal{U}_{q_
0\,q_1}$ and for the topological entropy of the associated subshift for ar
bitrary $q_0\, q_1 > 1$. We will also establish the continuity of these qu
antities as functions of the pair $(q_0\,q_1)$.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vítězslav Kala (Charles University)
DTSTART:20260407T120000Z
DTEND:20260407T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/165
DESCRIPTION:Title: Using geometric continued fractions for universal quadratic forms\nby Vítězslav Kala (Charles University) as part of One World Numeratio
n seminar\n\n\nAbstract\nAlready in the 18th century\, Lagrange proved tha
t every positive integer can be expressed as the sum of four squares of in
tegers. Nowadays we say that the sum of four squares is a universal quadra
tic form. In the friendly and accessible talk\, I’ll discuss some result
s on universal quadratic forms over Z and over totally real number fields.
In particular\, I'll focus on the role of continued fractions in many of
the advances in the area over the last 10 years. First\, I'll talk about t
he connection between classical continued fractions and quadratic forms ov
er real quadratic fields. Then I'll turn to the more complicated situation
of higher degree fields where we use geometric continued fractions as dev
eloped by Klein\, Arnold\, Karpenkov and many others. The talk is based on
recent joint works with Valentin Blomer\, Siu Hang Man\, Magdalena Tinkov
a\, Robin Visser\, and Pavlo Yatsyna.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jamie Walton (University of Nottingham)
DTSTART:20260421T120000Z
DTEND:20260421T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/166
DESCRIPTION:by Jamie Walton (University of Nottingham) as part of One Worl
d Numeration seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gandhar Joshi (The Open University)
DTSTART:20260505T120000Z
DTEND:20260505T130000Z
DTSTAMP:20260404T131146Z
UID:OWNS/167
DESCRIPTION:Title: Hiccup sequences\, a Kimberling's conjecture\, and Dumont-Thomas num
eration systems\nby Gandhar Joshi (The Open University) as part of One
World Numeration seminar\n\n\nAbstract\nThis is part of a joint work with
Robbert Fokkink (TU Delft). We generalise the self-referential sequences
introduced by Benoit Cloitre in 2003 on OEIS under the umbrella term ‘hi
ccup sequences’ with a direct skeletal influence from a ‘remarkable’
paper by Dekking\, Bosma\, and Steiner (2018) describing one such sequenc
e in five very interesting ways. In this talk\, we begin with one such way
that uses morphisms. Using the Dumont-Thomas numeration system (DTNS) ass
ociated with the morphism\, we prove a Kimberling’s conjecture on the OE
IS about the bounds of a difference sequence related to one such sequence.
There is also some use of Walnut software for Ollinger provides a tool th
at converts the DTNS into a set of Walnut-readable automata.\n
LOCATION:https://stable.researchseminars.org/talk/OWNS/167/
END:VEVENT
END:VCALENDAR