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BEGIN:VEVENT
SUMMARY:Osama Khalil (The University of Utah)
DTSTART:20200603T140000Z
DTEND:20200603T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/1
DESCRIPTION:Title: Random walks\, spectral gaps\, and Khintchine's theorem on fra
ctals\nby Osama Khalil (The University of Utah) as part of Webinar on
Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nIn 1984\
, Mahler asked how well typical points on Cantor’s set can be approximat
ed by rational numbers. His question fits within a program\, set out by hi
mself in the 1930s\, attempting to determine conditions under which subset
s of $\\mathbb{R}^d$ inherit the Diophantine properties of the ambient spa
ce. Since the approximability of typical points in Euclidean space by rati
onal points is governed by Khintchine’s classical theorem\, the ultimate
form of Mahler’s question asks whether an analogous zero-one law holds
for fractal measures. Significant progress has been achieved in recent yea
rs\, albeit\, almost all known results have been of “convergence type”
.\nIn this talk\, we will discuss the first instances where a complete ana
logue of Khinchine’s theorem for fractal measures is obtained. The class
of fractals for which our results hold includes those generated by ration
al similarities of $\\mathbb{R}^d$ and having sufficiently small Hausdorff
co-dimension. The main new ingredient is an effective equidistribution th
eorem for certain fractal measures on the space of unimodular lattices. Th
e latter is established via a new technique involving the construction of
$S$-arithmetic Markov operators possessing a spectral gap and encoding the
arithmetic structure of the maps generating the fractal. This is joint wo
rk in progress with Manuel Luethi.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Barak Weiss (Tel Aviv University)
DTSTART:20200610T123000Z
DTEND:20200610T140000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/2
DESCRIPTION:Title: New bounds on the covering density of a lattice\nby Barak
Weiss (Tel Aviv University) as part of Webinar on Diophantine approximatio
n and homogeneous dynamics\n\n\nAbstract\nWe obtain new upper bounds on th
e minimal density of lattice coverings of $\\mathbb{R}^n$ by dilates of a
convex body $K$. We also obtain bounds on the probability (with respect to
the natural Haar-Siegel measure on the space of lattices) that a randomly
chosen lattice $L$ satisfies $L+K=\\mathbb{R}^n$. Joint work with Or Orde
ntlich and Oded Regev.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (University of Oxford)
DTSTART:20200617T130000Z
DTEND:20200617T150000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/3
DESCRIPTION:Title: On the Duffin-Schaeffer Conjecture\nby James Maynard (Univ
ersity of Oxford) as part of Webinar on Diophantine approximation and homo
geneous dynamics\n\n\nAbstract\nAlmost 80 years ago Duffin and Schaeffer c
onjectured a beautiful strengthening of Khinchin's classical result: Given
a sequence of possible forms of rational approximation\, either almost al
l reals can be approximated in this manner or almost none can be\, and the
re is a simple calculation to tell which case we are in.\nI'll talk about
recent work with D. Koukoulopoulos which establishes this conjecture. This
relies on a blend of different techniques\, recasting the problem as a st
ructural question in additive combinatorics\, and then approaching this vi
a studying a particular family of graphs to reduce it to a problem in anal
ytic number theory.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lei Yang (Sichuan University)
DTSTART:20200701T133000Z
DTEND:20200701T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/4
DESCRIPTION:Title: Winning property of badly approximable points on curves\nb
y Lei Yang (Sichuan University) as part of Webinar on Diophantine approxim
ation and homogeneous dynamics\n\n\nAbstract\nWe will prove that badly app
roximable points (no matter weighted or unweighted) on any analytic non-de
generate curve in $\\mathbb{R}^n$ is an absolute winning set. This confirm
s a key conjecture in the area stated by Badziahin and Velani (2014) which
represents a far-reaching generalisation of Davenport's problem from the
1960s. Amongst various consequences of our main result is a solution to Bu
geaud's problem on real numbers badly approximable by algebraic numbers of
arbitrary degree. This work is joint with Victor Beresnevich and Erez Nes
harim.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nikolay Moshchevitin (Moscow State University)
DTSTART:20200708T133000Z
DTEND:20200708T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/5
DESCRIPTION:Title: Singular vectors: from Khintchine to nowadays\nby Nikolay
Moshchevitin (Moscow State University) as part of Webinar on Diophantine a
pproximation and homogeneous dynamics\n\n\nAbstract\nMany concepts in Diop
hantine Approximation have their origin in the \nfamous paper "Über eine
Klasse linearer diophantischer Approximationen" by A. Khintchine (1926). T
he results of this paper were rediscovered many times by different mathema
ticians. In particular\, Khintchine was the first who observed the phenome
non of singularity in higher-dimensional Diophantine Approximation. In our
lecture we discuss several problems related to singular vectors and best
approximation (minimal points) as well as some related topics dealing with
Diophantine exponents and approximation on algebraic and analytic surface
s which were considered recently in author's joint papers with D. Kleinboc
k and B. Weiss. Also we suppose to discuss some related open problems.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Baowei Wang (Huazhong University of Sci. & Tech.)
DTSTART:20200722T133000Z
DTEND:20200722T143000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/6
DESCRIPTION:Title: Mass transference principle from rectangles to rectangles in D
iophantine approximation\nby Baowei Wang (Huazhong University of Sci.
& Tech.) as part of Webinar on Diophantine approximation and homogeneous d
ynamics\n\n\nAbstract\nAs is well known\, Dirichlet's theorem and Minkowsk
i's theorem are two fundamental results in Diophantine approximation. One
says that all points in R^d will fall into infinitely many balls centered
at rationals with specific radius\; while the other says that all points w
ill fall into infinitely many rectangles centered at rationals with specif
ic sidelengths. This motives a further study on the metric theory of limsu
p sets defined by a sequence of balls or rectangles. Since the landmark w
orks of Beresnevich & Velani (2006) and Beresnevich\, Dickinson & Velani (
2006) where the mass transference principle was found\, the metric theory
for limsup sets defined by a sequence balls or isotropic thicken of genera
l sets has been sufficiently well established. While\, the metric theory f
or limsup sets defined by a sequence of rectangles are not as rich as the
ball case. In this talk\, I will talk about some progess on the metric the
ory of the latter case by modifying the settings in the above mentioned im
pressing works.\n\nThis talk will only take one hour.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengyu Yang (ETH Zurich)
DTSTART:20200729T133000Z
DTEND:20200729T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/7
DESCRIPTION:Title: Dirichlet Improvability\, Equidistribution\, and Grassmannians
\nby Pengyu Yang (ETH Zurich) as part of Webinar on Diophantine approx
imation and homogeneous dynamics\n\n\nAbstract\nAs a natural generalisatio
n of Dirichlet's approximation theorem on real numbers\, Dirichlet's appro
ximation theorem on $m\\times n$ real matrices tells us the following: giv
en $m$ real linear forms in $n$ variables\, we can find an integral vector
such that the evaluations of all the linear forms at this integral vector
are simultaneous small. In the 1960s Davenport and Schmidt showed that Di
richlet’s theorem is non-improvable for almost all matrices\, and they a
sked if the analogous result holds for a submanifold of the space of $m\\t
imes n$ matrices. This problem is related to an equidistribution problem i
n the space of unimodular lattices in $\\mathbb{R}^n$. In this talk I will
present some recent progress on this problem\, and I will explain its con
nections to the geometry of Grassmannian manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anurag Rao (Brandeis University)
DTSTART:20200805T130000Z
DTEND:20200805T140000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/8
DESCRIPTION:Title: Some problems in uniform Diophantine approximation\nby Anu
rag Rao (Brandeis University) as part of Webinar on Diophantine approximat
ion and homogeneous dynamics\n\n\nAbstract\nWe study a norm sensitive Diop
hantine approximation problem arising from the work of Davenport and Schmi
dt on the improvement of Dirichlet's theorem. Its supremum norm case was r
ecently considered by the Kleinbock and Wadleigh\, and here we extend the
set-up by replacing the supremum norm with an arbitrary norm. This gives r
ise to a class of shrinking target problems for one-parameter diagonal flo
ws on the space of lattices\, with the targets being neighborhoods of the
critical locus of a suitably scaled norm ball. We use methods from geometr
y of numbers and dynamics to generalize a result due to Andersen and Duke
on measure zero and uncountability of the set of numbers for which Minkows
ki approximation theorem can be improved. The choice of the Euclidean norm
on $\\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surf
ace which visit a decreasing family of balls. An application of a dynamica
l Borel-Cantelli lemma of Maucourant produces a zero-one law for improveme
nt of Dirichlet's theorem in Euclidean norm. Based on joint works with Dmi
try Kleinbock and Srinivasan Sathiamurthy.\n\nThe starting time is earlier
than usual. The talk will only take one hour.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anish Ghosh (Tata Institute of Fundamental Research)
DTSTART:20200812T133000Z
DTEND:20200812T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/9
DESCRIPTION:Title: Diophantine approximations\, large intersections and geodesics
in negative curvature\nby Anish Ghosh (Tata Institute of Fundamental
Research) as part of Webinar on Diophantine approximation and homogeneous
dynamics\n\n\nAbstract\nI will discuss new results on the `shrinking tar
get problem' including a logarithm law for approximation by geodesics in
negatively curved manifolds and Hausdorff dimension estimates for finer s
piraling phenomena of geodesics. I will also discuss the large intersectio
n property of Falconer in the context of negative curvature and some appli
cations to Diophantine approximation and to hyperbolic geometry. This is j
oint work with Debanjan Nandi.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Denis Koleda (Institute of mathematics\, Minsk)
DTSTART:20200819T133000Z
DTEND:20200819T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/10
DESCRIPTION:Title: The distribution of conjugate algebraic numbers: a random pol
ynomial approach\nby Denis Koleda (Institute of mathematics\, Minsk) a
s part of Webinar on Diophantine approximation and homogeneous dynamics\n\
n\nAbstract\nIn the talk we consider the spatial distribution of points th
at have algebraic (Galois) conjugate coordinates of fixed degree and bound
ed height. We give an asymptotic formula for counting such points in a wid
e class of regions of Euclidean space (as the parameter that bounds height
s grows to infinity). We explain connection of this formula to random poly
nomials with i.i.d. coefficients. We also discuss some corollaries and app
lications of the formula. The talk is based on a joint work with F. Götze
and D. Zaporozhets.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART:20200826T133000Z
DTEND:20200826T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/11
DESCRIPTION:Title: Higher order uniformity of the Liouville function\nby Kai
sa Matomäki (University of Turku) as part of Webinar on Diophantine appro
ximation and homogeneous dynamics\n\n\nAbstract\nThe Liouville function ta
kes a value +1 or -1 at a natural number $n$ depending on whether $n$ has
an even or an odd number of prime factors. The Liouville function is belie
ved to behave more or less randomly. In particular a famous conjecture of
Sarnak says that the Liouville function does not correlate with any sequen
ce of "low complexity" whereas a longstanding conjecture of Chowla says th
at the Liouville function has negligible correlations with its own shifts.
\nI will discuss conjectures of Sarnak and Chowla and my very recent work
with Radziwiłł\, Tao\, Teräväinen\, and Ziegler\, where we show that\,
in almost all intervals of length $X^\\varepsilon$\, the Liouville functi
on does not correlate with polynomial phases or more generally with nilseq
uences. I will also discuss applications to superpolynomial word complexit
y for the Liouville sequence and to a new averaged version of Chowla's con
jecture.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damien Roy (University of Ottawa)
DTSTART:20200923T133000Z
DTEND:20200923T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/12
DESCRIPTION:Title: Simultaneous rational approximation to exponentials of algebr
aic numbers\nby Damien Roy (University of Ottawa) as part of Webinar o
n Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nThe th
eorem of Lindemann-Weierstrass asserts that the exponentials of distinct a
lgebraic numbers are linearly independent over the field of rational numbe
rs. The proof uses a construction of simultaneous rational approximations
to such exponentials values\, which goes back to Hermite. In this talk\,
we show that\, from an adelic perspective\, these approximations are esse
ntially best possible. This point of view partly explains the nature of
the algebraic numbers whose exponentials have a structured continuous frac
tion expansion. We also propose few specific conjectures regarding sim
ultaneous approximations to such values in adèle rings.\n\nThe proof of o
ur main result requires a separate analysis for each place of the associat
ed number field. For the Archimedean places\, it relies on the structure
of the graph drawn in the complex plane by the paths of fastest descent f
or the norm of a general univariate complex polynomial starting from the r
oots of its derivative and ending in the roots of the polynomial. It h
appens that this graph is a tree and that the lengths of those paths can b
e estimated from above in terms of the degree of the given polynomial and
the diameter of its set of zeros. We will mention some instances where t
hese upper bounds can be greatly improved and state as an open problem whe
ther or not such improvements hold in general.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART:20200916T133000Z
DTEND:20200916T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/13
DESCRIPTION:Title: A fully-inhomogeneous version of Gallagher's theorem\nby
Sam Chow (University of Warwick) as part of Webinar on Diophantine approxi
mation and homogeneous dynamics\n\n\nAbstract\nGallagher's theorem describ
es the multiplicative diophantine \napproximation rate of a typical vector
. We establish a fully-inhomogeneous \nversion of Gallagher's theorem\, a
diophantine fibre refinement\, and a \nsharp and unexpected threshold for
Liouville fibres. Along the way\, we \nprove an inhomogeneous version of t
he Duffin--Schaeffer conjecture for a \nclass of non-monotonic approximati
on functions. Joint with Niclas Technau.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ralf Spatzier (University of Michigan)
DTSTART:20200930T133000Z
DTEND:20200930T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/14
DESCRIPTION:Title: Hyperbolic Actions of Higher Rank Abelian Groups\nby Ralf
Spatzier (University of Michigan) as part of Webinar on Diophantine appro
ximation and homogeneous dynamics\n\n\nAbstract\nWe study transitive $\\ma
thbb{R} ^k \\times \\mathbb{Z}^\\ell$ actions on arbitrary compact manifol
ds with a projectively dense set of Anosov elements and 1-dimensional coar
se Lyapunov foliations. Such actions are called totally Cartan actions. We
completely classify such actions as built from low-dimensional Anosov flo
ws and diffeomorphisms and affine actions\, verifying the Katok-Spatzier c
onjecture for this class. This is achieved by introducing a new tool\, the
action of a dynamically defined topological group describing paths in coa
rse Lyapunov foliations\, and understanding its generators and relations.
We obtain applications to the Zimmer program. This talk is based on joint
work with Kurt Vinhage.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mishel Skenderi (Brandeis University)
DTSTART:20201014T133000Z
DTEND:20201014T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/15
DESCRIPTION:Title: Small values at integer points of generic subhomogeneous func
tions\nby Mishel Skenderi (Brandeis University) as part of Webinar on
Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nThis tal
k will be based on joint work with Dmitry Kleinbock that has been motivate
d by several recent papers (among them\, those of Athreya-Margulis\, Bourg
ain\, Ghosh-Gorodnik-Nevo\, Kelmer-Yu). Given a certain sort of group $G$
and certain sorts of functions $f: \\mathbb{R}^n \\to \\mathbb{R}$ and $\\
psi : \\mathbb{R}^n \\to \\mathbb{R}_{>0}$\, we obtain necessary and suffi
cient conditions so that for Haar-almost every $g \\in G$\, there exist in
finitely many (respectively\, finitely many) $v \\in \\mathbb{Z}^n$ for wh
ich $|(f \\circ g)(v)| \\leq \\psi(\\|v\\|)$\, where $\\|\\cdot\\|$ is an
arbitrary norm on $\\mathbb{R}^n$. We also give a sufficient condition in
the setting of uniform approximation. As a consequence of our methods\, we
obtain generalizations to the case of vector-valued (simultaneous) approx
imation with no additional effort. In our work\, we use probabilistic resu
lts in the geometry of numbers that go back several decades to the work of
Siegel\, Rogers\, and W. Schmidt\; these results have recently found new
life thanks to a 2009 paper of Athreya-Margulis.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jialun Li (University of Zurich)
DTSTART:20201021T133000Z
DTEND:20201021T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/16
DESCRIPTION:Title: Decay of Fourier transforms of fractal measures\nby Jialu
n Li (University of Zurich) as part of Webinar on Diophantine approximatio
n and homogeneous dynamics\n\n\nAbstract\nWe will talk about some of the r
ecent works on estimating the decay of Fourier transforms of fractal measu
res\, such as self-similar measures and Furstenberg measures. The proof is
based on renewal theorems for stopping times of random walks on $\\mathbb
{R}$.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Pezzoni (University of York)
DTSTART:20201028T143000Z
DTEND:20201028T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/17
DESCRIPTION:Title: Simultaneous Diophantine approximation on manifolds by algebr
aic numbers\nby Alessandro Pezzoni (University of York) as part of Web
inar on Diophantine approximation and homogeneous dynamics\n\n\nAbstract\n
Simultaneous Diophantine approximation on manifolds is notoriously\ncompli
cated\, since it requires to take into account the arithmetic\nproperties
of a manifold\, as well as the analytic ones. In this talk\nwe will make s
ome progress towards a metric theory of approximation on\nmanifolds by alg
ebraic numbers with algebraic conjugate coordinates\,\ngeneralising a conj
ecture of Sprindžuk's.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Marnat (TU Graz)
DTSTART:20201104T143000Z
DTEND:20201104T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/18
DESCRIPTION:Title: Dirichlet is not just Bad and Sing\nby Antoine Marnat (TU
Graz) as part of Webinar on Diophantine approximation and homogeneous dyn
amics\n\n\nAbstract\nIt is well known that in dimension one\, the set of D
irichlet \nimprovable real numbers consists precisely of badly approximabl
e and \nsingular numbers. We show that in higher dimensions this disjoint
union \nis not the full set of Dirichlet improvable vectors: we prove that
there \nexist uncountably many Dirichlet improvable vectors that are neit
her \nbadly approximable nor singular. We construct these numbers using th
e \nparametric geometry of numbers. Furthermore\, by doing so we can choos
e \nthe exponent of Diophantine approximation by a rational subspace of \n
dimension exactly $d$\, for any d between $0$ and $n-1$.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Haynes (University of Houston)
DTSTART:20201111T133000Z
DTEND:20201111T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/19
DESCRIPTION:Title: Higher dimensional gap theorems in Diophantine approximation<
/a>\nby Alan Haynes (University of Houston) as part of Webinar on Diophant
ine approximation and homogeneous dynamics\n\n\nAbstract\nThe three distan
ce theorem states that\, if $x$ is any real number and $N$ is any positive
integer\, the points $x\, 2x\, … \, Nx \\mod 1$ partition the unit inte
rval into component intervals having at most $3$ distinct lengths. There a
re many higher dimensional analogues of this theorem\, and in this talk we
will discuss two of them. In the first we consider points of the form $mx
+ny \\mod 1$\, where $x$ and $y$ are real numbers and $m$ and $n$ are inte
gers taken from an expanding set in the plane. This version of the problem
was previously studied by Geelen and Simpson\, Chevallier\, Erdős\, and
many other people\, and it is closely related to the Littlewood conjecture
in Diophantine approximation. The second version of the problem is a stra
ightforward generalization to rotations on higher dimensional tori which\,
surprisingly\, has been largely overlooked in the literature. For the two
dimensional torus\, we are able to prove a five distance theorem\, which
is best possible. In higher dimensions we also have bounds\, but establish
ing optimal bounds is an open problem. The first hour of this talk will be
expository\, and the second half will focus on proofs. The new results pr
esented in this talk are joint work with Jens Marklof and with Roland Roed
er.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Bersudsky (IIT Technion)
DTSTART:20201118T143000Z
DTEND:20201118T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/20
DESCRIPTION:Title: On the image in the torus of sparse points on expanding analy
tic curves\nby Michael Bersudsky (IIT Technion) as part of Webinar on
Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nIt is kn
own that the projection to the $2$-torus of the normalised parameter measu
re on a circle of radius $R$ in the plane becomes uniformly distributed as
$R$ grows to infinity. I will discuss the following natural discrete anal
ogue for this problem. Starting from an angle and a sequence of radii $\\{
R_n\\}$ which diverges to infinity\, I will consider the projection to the
2-torus of the $n$'th roots of unity rotated by this angle and dilated by
a factor of $R_n$. The interesting regime in this problem is when $R_n$ i
s much larger than $n$ so that the dilated roots of unity appear sparsely
on the dilated circle. I will discuss 3 types of results:\n\n1. Validity o
f equidistribution for all angles when the sparsity is polynomial.\n\n2. F
ailure of equidistribution for some super polynomial dilations.\n\n3. Equi
distribution for almost all angles for arbitrary dilations.\n\nI will then
pass to discuss more general results on the projection to the $d$-torus o
f dilations of varying analytic curves in $d$-space.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Han Yu (University of Cambridge)
DTSTART:20201125T143000Z
DTEND:20201125T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/21
DESCRIPTION:Title: Rational points near self-similar sets\nby Han Yu (Univer
sity of Cambridge) as part of Webinar on Diophantine approximation and hom
ogeneous dynamics\n\n\nAbstract\nWe show that the rational points are quit
e well\n'equidistributed' near the middle 15th Cantor set $K$. As a conseq
uence\,\nit is possible to show that the set of well-approximable numbers
has\nfull Hausdorff dimension inside $K$. This answers a question of\nLeve
sley-Salp-Velani for $K$. In fact\, it is possible to prove a\nslightly st
ronger result which partially answers a question of\nBugeaud-Durand. The r
esults also hold for some self-similar sets other\nthan $K$. We will provi
de a sufficient condition and some other\nexamples. We suspect that the ab
ove results hold for all self-similar\nsets with Hausdorff dimension bigge
r than $1/2$ and with the open set\ncondition. We will see some heuristics
in the talk.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreyasi Datta (University of Michigan)
DTSTART:20201202T143000Z
DTEND:20201202T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/22
DESCRIPTION:Title: Recent progress in p-adic Diophantine approximation\nby S
hreyasi Datta (University of Michigan) as part of Webinar on Diophantine a
pproximation and homogeneous dynamics\n\n\nAbstract\nStudying the $p$-adic
analogue of Mahler's conjecture was initiated by Sprind zuk in 1969. Subs
equently\, there were several partial results culminating in the work of K
leinbock and Tomanov\, where the $S$-adic case of the Baker-Sprindzuk conj
ectures were settled in full generality. We provide a complete $p$-adic an
alogue of the results of D. Kleinbock on Diophantine exponents of affine s
ubspaces. This answers a conjecture of Kleinbock and Tomanov. Recently\, w
e proved $S$-arithmetic inhomogeneous Khintchine type theorems on analytic
nondegenerate manifolds. For $S$ consisting of more than one valuation\,
the divergence results are new even in the homogeneous setting. This aform
entioned result answers questions posed by Badziahin\, Beresnevich and Vel
ani and also it generalizes the work of Golsefidy and Mohammadi. In the fi
rst half of the talk\, I will go over these results and in the seocnd half
\, I will try to concentrate on some of the technical details of the proof
s. The new results presented in this talk are joint work with Anish Ghosh.
\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahriar Mirzadeh (Michigan State University)
DTSTART:20201209T143000Z
DTEND:20201209T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/23
DESCRIPTION:Title: Upper bound for the Hausdorff dimension of exceptional orbits
in homogeneous spaces\nby Shahriar Mirzadeh (Michigan State Universit
y) as part of Webinar on Diophantine approximation and homogeneous dynamic
s\n\n\nAbstract\nConsider the set of points in a homogeneous space $G/\\Ga
mma$\nwhose $g_t$-orbit misses a fixed open set. It has measure zero\nif t
he flow is ergodic. It has been conjectured that this set has\nHausdorff d
imension strictly smaller than the dimension of whole space. This\nconject
ure is proved when $G/\\Gamma$ is compact or when has real rank. In\nthis
talk we will prove the conjecture for probably the most important\nexample
of the higher rank case namely: $\\SL(m+n\, \\R)/\\SL(m+n\, \\Z)$ and $g_
t = \\mathrm{diag}\\{e^{t/m}\, \\dots \, e^{t/m}\, e^{-t/n}\, \\dots\, e^{
-t/n}\\}$. This\nhomogeneous space has many applications in Diophantine\na
pproximation that will be discussed in the talk if time permits. This\npro
ject is joint work with Dmitry Kleinbock.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Jitomirskaya (UC Irvine)
DTSTART:20201216T160000Z
DTEND:20201216T180000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/24
DESCRIPTION:Title: Inhomogeneous Diophantine approximation in the coprime settin
g\nby Svetlana Jitomirskaya (UC Irvine) as part of Webinar on Diophant
ine approximation and homogeneous dynamics\n\n\nAbstract\nGiven $n\\in\n$
and $x\\in\\R$\, let \n $$||nx||^\\prime=\\min\\{|nx-m|:m\\in\\Z\, gcd(n
\,m)=1\\}.$$\nTwo conjectures in the coprime inhomogeneous Diophantine app
roximation stated\, by analogy with the classical Diophantine approximatio
n\, that for any irrational number $\\alpha$ and almost every $\\gamma\\in
\\R$\,\n $$\\liminf_{n\\to \\infty}n||\\gamma -n\\alpha||^{\\prime}=0\,$$
\nand that there exists $C$ such that for all $\\gamma\\in \\R$\,\n\n$$\\l
iminf_{n\\to \\infty}n||\\gamma -n\\alpha||^{\\prime} < C.$$\n\nWe will pr
esent our joint work with W. Liu that proves one of those and disproves th
e other.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manos Zafeiropoulos (TU Graz)
DTSTART:20210210T143000Z
DTEND:20210210T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/25
DESCRIPTION:Title: Inhomogeneous Diophantine Approximation on $M_0$ Sets with re
stricted denominators\nby Manos Zafeiropoulos (TU Graz) as part of Web
inar on Diophantine approximation and homogeneous dynamics\n\n\nAbstract\n
Let $\\mu$ be a probability measure with $\\widehat{\\mu}(t)\\ll (\\log |t
|)^{-A}$ for some $A>0$\, supported on a set $F\\subseteq [0\,1]$. Let $\\
mathcal{A}=(q_n)_{n=1}^{\\infty}$ be an increasing \nsequence of intege
rs. We establish a quantitative inhomogeneous Khintchine-type theorem in
which the points of interest lie in $F$ and the "denominators" of the appr
oximants belong to $\\mathcal{A}$ in the following cases: \n(i) $(q_n)_
{n=1}^{\\infty}$ is lacunary and $A>2$.\n(ii)The prime divisors of $(q_n)_
{n=1}^{\\infty}$ are restricted in a set of $k$ prime numbers and $A>2k$.\
n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (Claude Bernard University Lyon 1)
DTSTART:20210217T143000Z
DTEND:20210217T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/26
DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite autom
ata\nby Boris Adamczewski (Claude Bernard University Lyon 1) as part o
f Webinar on Diophantine approximation and homogeneous dynamics\n\n\nAbstr
act\nIt is commonly expected that expansions of numbers in multiplicativel
y independent bases\, such as 2 and 10\, should have no common structure.
However\, it seems extraordinarily difficult to confirm this naive heurist
ic principle in some way or another. In the late 1960s\, Furstenberg sugge
sted a series of conjectures\, which became famous and aim to capture this
heuristic. The work I will discuss in this talk is motivated by one of th
ese conjectures. Despite recent remarkable progress by Shmerkin and Wu\, i
t remains totally out of reach of the current methods. While Furstenberg
’s conjectures take place in a dynamical setting\, I will use instead th
e language of automata theory to formulate some related conjectures that f
ormalize and express in a different way the same general heuristic. I will
explain how the latter follow from some recent advances in Mahler's metho
d\; a method in transcendental number theory initiated by Mahler in the en
d of the 1920s. This a joint work with Colin Faverjon.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Badziahin (University of Sydney)
DTSTART:20210224T100000Z
DTEND:20210224T120000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/27
DESCRIPTION:Title: Diophantine approximation on the Veronese curve\nby Dmitr
y Badziahin (University of Sydney) as part of Webinar on Diophantine appro
ximation and homogeneous dynamics\n\n\nAbstract\nIn the talk we discuss th
e set $S_n(\\lambda)$ of simultaneously $\\lambda$-well approximable point
s in $\\mathbb{R}^n$. These are the points $x$ such that the inequality\n$
$\\| x - p/q\\|_\\infty < q^{-\\lambda - \\epsilon}$$\nhas infinitely many
solutions in rational points $p/q$. Investigating the intersection of thi
s set with a suitable manifold comprises one of the most challenging probl
ems in Diophantine approximation. It is known that the structure of this s
et\, especially for large $\\lambda$\, depends on the manifold. For some m
anifolds it may be empty\, while for others it may have relatively large H
ausdorff dimension.\n\nWe will concentrate on the case of the Veronese cu
rve $V_n$. We discuss\, what is known about the Hausdorff dimension of the
set $S_n(\\lambda) \\cap V_n$ and will talk about the recent results of t
he speaker and Bugeaud which impose new bounds on that dimension.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minju Lee (Yale University)
DTSTART:20210317T143000Z
DTEND:20210317T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/28
DESCRIPTION:Title: Orbit closures of unipotent flows for hyperbolic manifolds wi
th Fuchsian ends.\nby Minju Lee (Yale University) as part of Webinar o
n Diophantine approximation and homogeneous dynamics\n\n\nAbstract\nThis i
s joint work with Hee Oh. We establish an analogue of Ratner's orbit closu
re theorem for any connected closed subgroup generated by unipotent elemen
ts in $\\mathrm{SO}(d\,1)$ acting on the space $\\Gamma\\backslash\\mathr
m{SO}(d\,1)$\, assuming that the associated hyperbolic manifold $M=\\Gamma
\\backslash\\mathbb{H}^d$ is a convex cocompact manifold with\nFuchsian en
ds. For $d = 3$\, this was proved earlier by McMullen\, Mohammadi and Oh.
In a higher dimensional case\, the possibility of accumulation on closed o
rbits of intermediate subgroups causes serious issues\, but in the end\, a
ll orbit closures of unipotent flows are relatively\nhomogeneous. Our resu
lts imply the following: for any $k\\geq 1$\,\n\n(1) the closure of any $k
$-horosphere in $M$ is a properly immersed submanifold\;\n\n(2) the closur
e of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold\
;\n\n(3) an infinite sequence of maximal properly immersed geodesic $(k+1)
$-planes intersecting $\\mathrm{core} M$ becomes dense in $M$.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pablo Shmerkin (Torcuato Di Tella University)
DTSTART:20210310T163000Z
DTEND:20210310T183000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/29
DESCRIPTION:Title: Beyond Furstenberg's intersection conjecture\nby Pablo Sh
merkin (Torcuato Di Tella University) as part of Webinar on Diophantine ap
proximation and homogeneous dynamics\n\n\nAbstract\nHillel Furstenberg con
jectured in the 1960s that the intersections of closed $\\times 2$ and $\\
times 3$-invariant Cantor sets have "small" Hausdorff dimension. This conj
ecture was proved independently by Meng Wu and by myself\; recently\, Tim
Austin found a simple proof. I will present some generalizations of the in
tersection conjecture and other related results.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lifan Guan (Gottingen)
DTSTART:20210414T133000Z
DTEND:20210414T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/30
DESCRIPTION:Title: Divergent trajectories on products of homogenous spaces\n
by Lifan Guan (Gottingen) as part of Webinar on Diophantine approximation
and homogeneous dynamics\n\n\nAbstract\nThanks to Dani correspondence\, it
is now well-known that the set of singular vectors is closely related to
the set of points with divergent trajectories in certain homogeneous dynam
ical systems. Since Yitwah Cheung's breakthrough work on the Hausdorff dim
ension of the set of 2-dim singular vectors\, there have been lots of prog
ress in singular vectors and divergent trajectories in the so-called "unwe
ighted" cases. Otherwise\, our understanding is quite limited. In this tal
k\, I will mainly discuss the dimension formula for the set of divergent t
rajectories in products of "unweighted" homogeneous dynamical systems. Thi
s is a joint work with Jinpeng An\, Antoine Marnat and Ronggang Shi.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Jurga (University of St Andrews)
DTSTART:20210421T133000Z
DTEND:20210421T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/31
DESCRIPTION:Title: Random matrix products and self-projective sets\nby Natal
ia Jurga (University of St Andrews) as part of Webinar on Diophantine appr
oximation and homogeneous dynamics\n\n\nAbstract\nA finite set of matrices
$A \\subset SL(2\,R)$ acts on one-dimensional real projective space $RP^1
$ through its linear action on $R^2$. In this talk we will be interested i
n the smallest closed subset of $RP^1$ which contains all attracting and n
eutral fixed points of matrices in $A$ and which is invariant under the pr
ojective action of $A$. Recently\, Solomyak and Takahashi proved that if $
A$ is uniformly hyperbolic and satisfies a Diophantine property\, then the
invariant set has Hausdorff dimension equal to the minimum of 1 and the c
ritical exponent. In this talk we will discuss an extension of their resul
t beyond the uniformly hyperbolic setting. This is based on joint work wit
h Argyrios Christodoulou.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pratyush Sarkar (Yale University)
DTSTART:20210428T133000Z
DTEND:20210428T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/32
DESCRIPTION:Title: Generalization of Selberg's 3⁄16 theorem for convex cocompa
ct thin subgroups of SO(n\, 1)\nby Pratyush Sarkar (Yale University) a
s part of Webinar on Diophantine approximation and homogeneous dynamics\n\
n\nAbstract\nSelberg’s 3/16 theorem for congruence covers of the modular
surface is a beautiful theorem which has a natural dynamical interpretati
on as uniform exponential mixing. Bourgain-Gamburd-Sarnak's breakthrough w
orks initiated many recent developments to generalize Selberg's theorem fo
r infinite volume hyperbolic manifolds. One such result is by Oh-Winter es
tablishing uniform exponential mixing for convex cocompact hyperbolic surf
aces. These are not only interesting in and of itself but can also be used
for a wide range of applications including uniform resonance free regions
for the resolvent of the Laplacian\, affine sieve\, and prime geodesic th
eorems. I will present a further generalization to higher dimensions and s
ome of these immediate consequences.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henna Koivusalo (University of Bristol)
DTSTART:20210505T133000Z
DTEND:20210505T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/33
DESCRIPTION:Title: Linear repetitivity in polytopal cut and project sets\nby
Henna Koivusalo (University of Bristol) as part of Webinar on Diophantine
approximation and homogeneous dynamics\n\n\nAbstract\nCut and project set
s are aperiodic point patterns obtained by projecting an irrational slice
of the integer lattice to a subspace. One way of classifying aperiodic set
s is to study the number and repetition of finite patterns. From this pers
pective\, sets with patterns repeating linearly often\, called linearly re
petitive sets\, can be viewed as the most ordered aperiodic sets. Repetiti
vity of a cut and project set depends on the slope and shape of the irrati
onal slice. The cross-section of the slice is known as the window. In earl
ier works\, joint with subsets of {Haynes\, Julien\, Sadun\, Walton}\, we
showed that many properties of cut and project sets with a cube window can
be studied in the language of Diophantine approximation. For example\, li
near repetitivity holds if and only if the following two conditions are sa
tisfied: (i) the cut and project set has minimal number of different finit
e patterns (minimal complexity)\, and (ii) the irrational slope satisfies
a badly approximable condition. In a new joint work with Jamie Walton\, we
give a generalisation of this result to all polytopal windows satisfying
a mild geometric condition. A key step in the proof is a decomposition of
the cut and project scheme\, which allows us to make sense of condition (i
i) for general polytopal windows.\nThe talk will cover motivation and hist
ory of studying cut and project sets\, showcase a series of results on the
ir repetitivity properties highlighting the number theory connections\, an
d finish with the new results which move beyond Diophantine approximation.
\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (University of Göttingen)
DTSTART:20210512T133000Z
DTEND:20210512T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/34
DESCRIPTION:Title: Density of rational points near/on compact manifolds with cer
tain curvature conditions\nby Damaris Schindler (University of Göttin
gen) as part of Webinar on Diophantine approximation and homogeneous dynam
ics\n\n\nAbstract\nIn this talk I will discuss joint work with Shuntaro Ya
magishi where we establish an asymptotic formula for the number of rationa
l points\, with bounded denominators\, within a given distance to a compac
t submanifold $M$ of $\\mathbb{R}^n$ with a certain curvature condition. T
echnically we build on work of Huang on the density of rational points nea
r hypersurfaces. One of our goals is to explore generalisations to higher
codimension. In particular we show that assuming certain curvature conditi
ons in codimension at least two\, leads to upper bounds for the number of
rational points on $M$ which are even stronger than what would be predicte
d by the analogue of Serre's dimension growth conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nattalie Tamam (UC San Diego)
DTSTART:20210519T153000Z
DTEND:20210519T173000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/35
DESCRIPTION:Title: When can we find a simple description of the divergence of tr
ajectories?\nby Nattalie Tamam (UC San Diego) as part of Webinar on Di
ophantine approximation and homogeneous dynamics\n\n\nAbstract\nIt is well
known that the only singular numbers are rational numbers. Dani's corresp
ondence ties this property to a simple algebraic description of divergent
trajectories in $\\mathrm{SL}_2(\\mathbb{R})$ under the action of the diag
onal group. Similar principles can be utilised to define obvious divergent
trajectories in a more general setting. We will discuss the existence of
non-obvious divergent trajectories under the action of different diagonal
subgroups\, and the diophantine meaning of their existence (or lack thereo
f). This is a joint work with Omri Nisan Solan.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Scoones (University of York)
DTSTART:20210526T133000Z
DTEND:20210526T153000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/36
DESCRIPTION:Title: On the abc Conjecture in Algebraic Number Fields\nby Andr
ew Scoones (University of York) as part of Webinar on Diophantine approxim
ation and homogeneous dynamics\n\n\nAbstract\nWhile the abc Conjecture rem
ains open\, much work has been done on weaker versions\, and on generalisi
ng the conjecture to number fields. Stewart and Yu were able to give an ex
ponential bound for $\\max\\{a\, b\, c\\}$ in terms of the radical over th
e integers\, while Györy was able to give an exponential bound for the pr
ojective height $H(a\, b\, c)$ in terms of the radical for algebraic integ
ers. We generalise Stewart and Yu's method to give an improvement on Györ
y's bound for algebraic integers.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Poels and Damien Roy (University of Ottawa)
DTSTART:20211008T140000Z
DTEND:20211008T160000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/37
DESCRIPTION:Title: Simultaneous rational approximation to successive powers of a
real number\nby Anthony Poels and Damien Roy (University of Ottawa) a
s part of Webinar on Diophantine approximation and homogeneous dynamics\n\
n\nAbstract\nLet $\\xi\\in\\mathbb{R}\\setminus\\bar{\\mathbb{Q}}$ be a re
al transcendental number and let $n$ be a positive integer. By pioneer wo
rk of Davenport \nand Schmidt from 1969\, we know that the exponent $\\tau
_{n+1}(\\xi)$ \nof best approximation to $\\xi$ by algebraic integers of d
egree\nat most $n+1$ is at least equal to $1+1/\\lambda_n(\\xi)$\, where \
n$\\lambda_n(\\xi)$ stands for the uniform exponent of rational\napproxima
tion to the successive powers $1\,\\xi\,\\dots\,\\xi^n$ \nof $\\xi$. So a
ny upper bound on $\\lambda_n(\\xi)$ which holds \nfor any $\\xi\\in\\math
bb{R}\\setminus\\bar{\\mathbb{Q}}$ provides a lower bound on \n$\\tau_{n+1
}(\\xi)$ which is also independent of $\\xi$. In this talk\, \nwe present
new tools which yield\, for each integer $n\\ge 4$\, a\nsignificantly imp
roved upper bound on $\\lambda_n(\\xi)$ and thus\na refined lower bound on
$\\tau_{n+1}(\\xi)$. The new lower bound is\n$n/2+a\\sqrt{n}+4/3$ with $
a=(1-\\log(2))/2\\simeq 0.153$\, instead of the\ncurrent $n/2+\\mathcal{O}
(1)$.\n\nAs usual\, the starting point is the sequence of so-called \nmini
mal points $\\mathbf{x}_1\,\\mathbf{x}_2\,\\mathbf{x}_3\,\\ldots$ in $\\ma
thbb{Z}^{n+1}$ defined initially \nby Davenport and Schmidt. Our strategy
consists in estimating from above \nthe height of the subspaces of $\\math
bb{R}^{n-\\ell+1}$ generated by $n-\\ell+1$ \nconsecutive coordinates from
each point among $\\mathbf{x}_i\,\\mathbf{x}_{i+1}\,\\dots\,\\mathbf{x}_q
$\nfor given $i\\le q$. To this end\, we first need a lower bound for the
\ndimension of such spaces.\n\nIn the first part of the talk\, we present
the required background with some historical perspective and our key alge
braic result concerning the dimension of the above mentioned spaces. In th
e second part\, we look at their heights from different \nperspectives and
outline the general strategy of the proof.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Gorodnik (University of Zurich)
DTSTART:20211022T140000Z
DTEND:20211022T160000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/38
DESCRIPTION:Title: Products of linear forms and distribution of translated measu
res\nby Alex Gorodnik (University of Zurich) as part of Webinar on Dio
phantine approximation and homogeneous dynamics\n\n\nAbstract\nWe explore
the behavior of the counting function which represents the number of solut
ions\nof a multiplicative Diophantine problem. The argument is based on an
alysis of measures translated under a group action on homogeneous spaces.
Ultimately we explain how estimates on correlations of translated measures
lead to a quantitative asymptotic formula for the counting function. This
is a joint work with Björklund and Fregolli.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jimmy Tseng (University of Exeter)
DTSTART:20211105T143000Z
DTEND:20211105T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/39
DESCRIPTION:Title: Shrinking target and horocycle equidistribution\nby Jimmy
Tseng (University of Exeter) as part of Webinar on Diophantine approximat
ion and homogeneous dynamics\n\n\nAbstract\nConsider a shrinking neighborh
ood of a cusp of the unit tangent bundle of a noncompact hyperbolic surfac
e of finite area. We discuss how a closed horocycle whose length goes t
o infinity can become equidistributed on this shrinking neighborhood\, giv
ing a sharp criterion in a natural case. This setup is closely related to
number theory\, and\, as an example\, our method yields a number-theoretic
identity.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yitwah Cheung (Tsinghua University)
DTSTART:20211119T143000Z
DTEND:20211119T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/40
DESCRIPTION:Title: Invariants of Diophantine Approximation\nby Yitwah Cheung
(Tsinghua University) as part of Webinar on Diophantine approximation and
homogeneous dynamics\n\n\nAbstract\nThere is a natural generalization of
the concept of convergents of the continued fraction to higher dimensions
that does not involve any specific choice of norm. In this talk\, I will
motivate this concept from several different angles\, within the framework
of staircases\, which is a rectilinear version of Kleinian sails. I will
describe some results about dual convergents and illustrate the method of
our approach towards constructing slowly unbounded A-orbits by sketching
the proof of dichotomy of Hausdorff dimension phenomenon obtained in joint
work with P. Hubert and H. Masur.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiyoung Han (Tata Institute)
DTSTART:20211203T143000Z
DTEND:20211203T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/41
DESCRIPTION:Title: Higher moment formulas for Siegel transforms and applications
to limit distributions of functions of counting lattice points\nby Ji
young Han (Tata Institute) as part of Webinar on Diophantine approximation
and homogeneous dynamics\n\n\nAbstract\nThe Siegel transform is one of th
e main tools when we consider\nproblems related to counting lattice points
using homogeneous dynamics. It\nis revealed by many mathematicians that S
iegel’s integral formula and\nRogers’ second moment formula are very u
seful to solve various\nquantitative and effective variants of classical p
roblems in the geometry\nof numbers\, such as the Gauss circle problem (ge
neralized to convex sets)\nand Oppenheim conjecture. Furthermore\, Rogers
’ higher moment formulas\,\ntogether with the method of moments\, give u
s information about limit\ndistributions related to these problems.\nIn th
is talk\, we revisit Rogers’ higher moment formulas with a new\napproach
\, and introduce higher moment formulas for Siegel transforms on\nthe spac
e of affine unimodular lattices and the space of unimodular\nlattices with
a congruence condition. Using these formulas\, we obtain the\nresults of
limit distributions\, which are generalizations of the work of\nRogers (19
56)\, Södergren (2011)\, and Strömbergsson and Södergren (2019).\nThis
is joint work with Mahbub Alam and Anish Ghosh.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Demi Allen (University of Warwick)
DTSTART:20211217T143000Z
DTEND:20211217T163000Z
DTSTAMP:20260404T111107Z
UID:DAHD-webinar/42
DESCRIPTION:Title: An inhomogeneous Khintchine-Groshev Theorem without monotonic
ity\nby Demi Allen (University of Warwick) as part of Webinar on Dioph
antine approximation and homogeneous dynamics\n\n\nAbstract\nThe classical
(inhomogeneous) Khintchine-Groshev Theorem tells us that for a monotonic
approximating function $\\psi: \\mathbb{N} \\to [0\,\\infty)$ the Lebesgue
measure of the set of (inhomogeneously) $\\psi$-well-approximable points
in $\\R^{nm}$ is zero or full depending on\, respectively\, the convergenc
e or divergence of $\\sum_{q=1}^{\\infty}{q^{n-1}\\psi(q)^m}$. In the homo
geneous case\, it is now known that the monotonicity condition on $\\psi$
can be removed whenever $nm>1$\, and cannot be removed when $nm=1$. In thi
s talk I will discuss recent work with Felipe A. Ramírez (Wesleyan\, US)
in which we show that the inhomogeneous Khintchine-Groshev Theorem is true
without the monotonicity assumption on $\\psi$ whenever $nm>2$. This resu
lt brings the inhomogeneous theory almost in line with the completed homog
eneous theory. I will survey previous results towards removing monotonicit
y from the homogeneous and inhomogeneous Khintchine-Groshev Theorem before
discussing the main ideas behind the proof our recent result.\n
LOCATION:https://stable.researchseminars.org/talk/DAHD-webinar/42/
END:VEVENT
END:VCALENDAR