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BEGIN:VEVENT
SUMMARY:Piotr Miska (Jagiellonian University in Krak\\'{o}w\, Poland)
DTSTART:20240524T130000Z
DTEND:20240524T132500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/1
DESCRIPTION:Title: On the Frobenius problem with restrictions on
common divisors of coefficients\nby Piotr Miska (Jagiellonian Universi
ty in Krak\\'{o}w\, Poland) as part of Combinatorial and additive number t
heory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY G
raduate Center.\n\nAbstract\nLet $m\,s\,t$ be positive integers with $t\\l
eq s-2$ and let $a_1\,a_2\,\\ldots\,a_s$ be positive integers such that $(
a_1\,a_2\,\\ldots\,a_{s-1})=1$. In the paper we prove that every sufficien
tly large positive integer can be written in the form $a_1\\mu_1+a_2\\mu_2
+\\ldots+a_s\\mu_m$\, where the positive integers $\\mu_1\,\\mu_2\,\\ldot
s\,\\mu_s$ have no common divisor that is the $m$-th power of a positive i
nteger greater than $1$\, but each $t$ of the values $\\mu_1\,\\mu_2\,\\l
dots\,\\mu_s$ do have a common divisor that is the $m$-th power of a posit
ive integer greater than $1$. Moreover\, we show that every sufficiently l
arge positive integer can be written as a sum of positive integers $\\mu_1
\,\\mu_2\,\\ldots\,\\mu_s$ with no common divisor that is the $m$-th power
of a positive integer greater than $1$\, but each $s-1$ of the values of
$\\mu_1\,\\mu_2\,\\ldots\,\\mu_s$ do have a common divisor that is the $m$
-th power of a positive integer greater than $1$. \n\nJoint work with Maci
ej Zakarczemny (Cracow University of Technology).\\\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:I.D. Shkredov (Purdue University)
DTSTART:20240524T133000Z
DTEND:20240524T135500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/2
DESCRIPTION:Title: On universal sets and sumsets\nby I.D. Shk
redov (Purdue University) as part of Combinatorial and additive number the
ory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Gra
duate Center.\n\nAbstract\nLet $G$ be an abelian group. A set $A \\subsete
q G$ is called \na {\\it $k$--universal} set if for any $x_1\,\\dots\,x_k
\\in G$ there exists $s\\in G$ \nsuch that $x_1+s\,\\dots\,x_k+s \\in A$.
The term ``universal set'' was introduced \nby Alon\, Bukh\, and Sudakov
in connection with the discrete Kakeya problem. \nWe study the concept of
universal sets from the additive--combinatorial point of view. Among othe
r results we obtain some applications of this type of uniformity to sets a
voiding solutions to linear equations\, and get an optimal upper bound for
the covering number of general sumsets.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Beker (University of Zagreb\, Croatia)
DTSTART:20240524T140000Z
DTEND:20240524T142500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/3
DESCRIPTION:Title: On a problem of Erdos and Graham about consecu
tive sums in strictly increasing sequences\nby Adrian Beker (Universit
y of Zagreb\, Croatia) as part of Combinatorial and additive number theory
(CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Gradua
te Center.\n\nAbstract\nGiven a finite sequence of integers $a = (a_i)_{1\
\leq i \\leq k}$\, let $S(a)$ denote the set of its consecutive sums\, tha
t is\, sums of the form $\\sum_{i=u}^{v}a_i$ with $1 \\leq u \\leq v \\leq
k$. Erd\\H os and Graham asked whether there exists a constant $c > 0$ s
uch that\, for all positive integers $n$\, there is such a sequence in $\\
{1\,\\ldots\,n\\}$ which is strictly increasing and satisfies $|S(a)| \\ge
q cn^2$. \n\nThe obvious candidate consisting of all integers from $1$ up
to $n$ falls short of having this property due to reasons related to the m
ultiplication table problem. On the other hand\, if we drop the monotonici
ty assumption\, such sequences were shown to exist by Hegyv\\'ari via a c
onstruction based on Sidon sets. In this talk\, I will present two constr
uctions\, one probabilistic and the other deterministic\, that give an aff
irmative answer to the starting question. I will also discuss some non-tr
ivial upper bounds on the size of $S(a)$ in this setting.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Chapman (University of Bristol\, UK)
DTSTART:20240524T143000Z
DTEND:20240524T145500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/4
DESCRIPTION:Title: Monochromatic sums and products\nby Jonath
an Chapman (University of Bristol\, UK) as part of Combinatorial and addit
ive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 i
n the CUNY Graduate Center.\n\nAbstract\nIf we colour $\\{2\,\\ldots\,N\\}
$ with $r$ different colours\, how many monochromatic solutions to $xy=z$
appear? A classical theorem of Schur shows that we always obtain $(c_r+o(1
))N^2$ monochromatic solutions (as $N\\to\\infty$) to $x+y=z$\, for some $
c_r>0$\, which is within a constant factor of the total number of solution
s. However\, Prendiville showed that one cannot achieve such a strong resu
lt for $xy=z$\, even if one only uses $2$ colours.\n \n In this talk\, I w
ill present recent work on determining the asymptotic minimum number of mo
nochromatic solutions to $xy=z$. We prove that every $2$-colouring of $\\{
2\,\\ldots\,N\\}$ produces \n at least $(2^{-3/2} + o(1))\\sqrt{N}\\log N$
monochromatic solutions to $xy=z$\, and the leading constant is sharp. I
will also introduce a Schur-type problem for colourings of real numbers. I
f the discrete and continuous Schur problems are `quantitatively equivalen
t'\, then\, for an arbitrary number of colours\, our upper and lower bound
s for the number of monochromatic solutions to $xy=z$ match up to a logari
thmic factor.\n \n Joint work with Lucas Aragao\, Miquel Ortega\, and Vict
or Souza.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrzej Kukla (Jagiellonian University in Krak\\'{o}w\, Poland)
DTSTART:20240524T150000Z
DTEND:20240524T152500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/5
DESCRIPTION:Title: Representing the number of binary partitions a
s sums of three squares\nby Andrzej Kukla (Jagiellonian University in
Krak\\'{o}w\, Poland) as part of Combinatorial and additive number theory
(CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Gradua
te Center.\n\nAbstract\nLet $c_m(n)$ denote the number of partitions of $n
$ into parts that are powers of 2 such that part equal to 1 takes one amon
g $2m$ colors and each part $>1$ takes one among $m$ colors. The study of
this function was initiated in 2021 by Zmija and Ulas\, who focused on its
2-adic behaviour. In particular\, they found a formula for the 2-adic val
uation of $c_m(n)$ that depends on the 2-adic valuation of $m$ and the val
ue of $t_n - t_{n-1}$\, where $t_n$ is the $n$-th term of the Prouhet-Thue
-Morse sequence. During the talk we will be considering the diophantine eq
uation $c_m(n) = x^2 + y^2 + z^2$. For fixed $m$\, we will characterize th
e set of natural numbers $n$\, for which the solution does not exist\, and
then further investigate properties of these sets. The talk is based on a
n ongoing work on speaker's master's thesis.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Technau (Paderborn University\, Germany)
DTSTART:20240524T153000Z
DTEND:20240524T155500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/6
DESCRIPTION:Title: Cilleruelo's conjecture on the LCM of polynomi
al sequences\nby Marc Technau (Paderborn University\, Germany) as part
of Combinatorial and additive number theory (CANT 2024)\n\nLecture held i
n Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nWe dis
cuss a conjecture of Cilleruelo on the growth of the least common multiple
of consecutive values of a polynomial and subsequent progress towards it
in work of Maynard--Rudnick and Sah.\nIn recent work\, the speaker and\, i
ndependently\, Alexei Entin made further advances by exploiting symmetries
amongst the roots of the polynomials in question.\nWe shall discuss these
approaches and related beautiful work of Baier and Dey.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bartosz Sobolewski (Jagiellonian University\, Krak\\'{o}w\, Poland
and Montanuniversit{\\"a}t Leoben\, Austria)
DTSTART:20240524T173000Z
DTEND:20240524T175500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/8
DESCRIPTION:Title: On block occurrences in the binary expansions
of $n$ and $n+t$\nby Bartosz Sobolewski (Jagiellonian University\, Kra
k\\'{o}w\, Poland and Montanuniversit{\\"a}t Leoben\, Austria) as part of
Combinatorial and additive number theory (CANT 2024)\n\nLecture held in Ro
om 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $s(n)$
denote the sum of binary digits of a nonnegative integer $n$. In the rece
nt years there has been significant progress concerning the behavior of th
e differences $s(n+t)-s(n)$\, where $t$ is a fixed nonnegative integer. In
particular\, Spiegelhofer and Wallner proved that for $t$ having sufficie
ntly many blocks $01$ in its binary expansion\, the set $\\{n: s(n+t) \\ge
q s(n)\\}$ has natural density $> 1/2$ (partially confirming a conjecture
by Cusick). Moreover\, for such $t$ the distribution $s(n+t) - s(n)$ is cl
ose to Gaussian. During the talk we consider an analogue of this problem c
oncerning the function $r(n)$\, which counts the occurrences of the block
$11$ in the binary expansion of $n$. In particular\, we prove that the di
stribution of $r(n+t)-r(n)$ is approximately Gaussian as well. We also dis
cuss a generalization to an arbitrary block of binary digits.\n\nJoint wor
k with Lukas Spiegelhofer (Montanuniversit{\\"a}t Leoben).\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Alfr\\' ed R\\' enyi Institute of Mathematics\, Buda
pest\, Hungary)
DTSTART:20240524T180000Z
DTEND:20240524T182500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/9
DESCRIPTION:Title: Solutions to the discrete Pompeiu problem and
to the finite Steinhaus tiling problem\nby Gergely Kiss (Alfr\\' ed R\
\' enyi Institute of Mathematics\, Budapest\, Hungary) as part of Combinat
orial and additive number theory (CANT 2024)\n\nLecture held in Room 4102
and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $K$ be a nonem
pty finite subset of the Euclidean space $\\mathbb{R}^k$ $(k\\ge 2)$.\nIn
this talk we discuss the solution of the following so-called discrete Pomp
eiu problem. If a function $f\\colon \\mathbb{R}^k \\to \\mathbb{C}$ is su
ch that the sum of $f$\non every congruent copy of $K$ is zero\, then $f$
vanishes everywhere. In fact\, we solve\na stronger\, weighted version of
this problem. As a corollary we obtain that every\nfinite subset of $\\mat
hbb{R}^k$ having at least two elements is a Jackson\nset\; that is\, no su
bset of $\\mathbb{R}^k$ intersects every congruent copy of $K$ in\nexactly
one point.\n\nJoint work with Mikl\\' os Laczkovich.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Cumberbatch (Purdue University)
DTSTART:20240524T183000Z
DTEND:20240524T185500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/10
DESCRIPTION:Title: Digitally restricted sets and the Goldbach co
njecture\nby James Cumberbatch (Purdue University) as part of Combinat
orial and additive number theory (CANT 2024)\n\nLecture held in Room 4102
and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nWe show that given
any base $b$ and any set of digits $\\mathcal{D}$ with at least two digit
s\, let $\\mathcal{A}$ be the set of integers whose base-$b$ digits consis
t only of values in $\\mathcal{D}$. We prove that almost all even integers
in $\\mathcal{A}$ are sum of two primes.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20240524T190000Z
DTEND:20240524T192500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/11
DESCRIPTION:Title: Polynomial equations in infinitely many varia
bles\nby Mel Nathanson (Lehman College (CUNY)) as part of Combinatoria
l and additive number theory (CANT 2024)\n\nLecture held in Room 4102 and
Room 9207 in the CUNY Graduate Center.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mizan Khan (Eastern Connecticut State University)
DTSTART:20240524T193000Z
DTEND:20240524T195500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/12
DESCRIPTION:Title: A conjecture for clean lattice parallelograms
\nby Mizan Khan (Eastern Connecticut State University) as part of Comb
inatorial and additive number theory (CANT 2024)\n\nLecture held in Room 4
102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $P \\subse
teq {\\mathbb R}^2$ be a convex lattice polygon containing at least one la
ttice point in its interior. The interior hull of $P$\, denoted by $P^{(1)
}$\, is the convex closure of the set of lattice points in the interior of
$P$\, that is\,\n$$P^{(1)} = \\operatorname{conv}\\left(\\operatorname{in
terior}(P) \\cap {\\mathbb Z}^2\\right).$$\nWe can now form a finite neste
d sequence of interior hulls\n$$P^{(1)}\\supseteq P^{(2)} \\supseteq P^{(3
)} \\supseteq \\ldots\, $$\nwhere $P^{(2)}$ is the interior hull of $P^{(1
)}$\, $P^{(3)}$ is the interior hull of $P^{(2)}$\, and so on. \n\nWe wil
l present some experimental data supporting a conjecture on the average nu
mber of interior hulls for clean lattice parallelograms. (A lattice paral
lelogram is said to be clean if the only lattice points on its boundary ar
e the 4 vertices.) \n\nJoint work with Riaz Khan.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathaniel Kingsbury (CUNY Graduate Center)
DTSTART:20240524T200000Z
DTEND:20240524T202500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/13
DESCRIPTION:Title: The square-root law does not hold in the pres
ence of zero divisors\nby Nathaniel Kingsbury (CUNY Graduate Center) a
s part of Combinatorial and additive number theory (CANT 2024)\n\nLecture
held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\n
Let $R$ be a finite ring and define the paraboloid $P = \\{(x_1\, \\dots\,
x_d)\\in R^d|x_d = x_1^2 + \\dots + x_{d-1}^2\\}.$ Suppose that for a seq
uence of finite rings of size tending to infinity\, the Fourier transform
of $P$ satisfies a square-root type bound constant $C$. Then all but finit
ely many of the rings are fields.\n\nMost of our argument works in greater
generality: let $f$ be a polynomial with integer coefficients in $d-1$ va
riables\, with a fixed order of variable multiplications (so that it defin
es a function $R^{d-1}\\rightarrow R$ even when $R$ is noncommutative)\, a
nd set $V_f = \\{(x_1\, \\dots\, x_d)\\in R^d|x_d = f(x_1\, \\dots\, x_{d
-1})\\}$. If (for a sequence of finite rings of size tending to infinity)
we have a square-root type bound on the Fourier transform of $V_f$\, then
all but finitely many of the rings are fields or matrix rings of small dim
ension.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20240524T203000Z
DTEND:20240524T205500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/14
DESCRIPTION:Title: Multi-parameter point-line incidence estimate
s in finite fields and applications\nby Steven Senger (Missouri State
University) as part of Combinatorial and additive number theory (CANT 2024
)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Center.\
n\nAbstract\nWe present some novel multi-parameter point-line incidence es
timates in vector spaces over finite fields. These outperform the straight
forward higher-dimensional analogs. We focus on an application to sums and
products type problems.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fred Tyrrell (University of Bristol\, UK)
DTSTART:20240522T153000Z
DTEND:20240522T155500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/15
DESCRIPTION:Title: New lower bounds for cap sets\nby Fred Ty
rrell (University of Bristol\, UK) as part of Combinatorial and additive n
umber theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the
CUNY Graduate Center.\n\nAbstract\nA cap set is a subset of $\\mathbb{F}_
3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. The cap set p
roblem asks how large a cap set can be\, and is an important problem in ad
ditive combinatorics and combinatorial number theory. In this talk\, I wil
l introduce the problem\, give some background and motivation\, and descri
be how I was able to provide the first progress in 20 years on the lower b
ound for the size of a maximal cap set. Building on a construction of Edel
\, we use improved computational methods and new theoretical ideas to show
that\, for large enough $n$\, there is always a cap set in $\\mathbb{F}_3
^n$ of size at least $2.218^n$. I will then also discuss recent developmen
ts\, including an extension of this result by Google DeepMind.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:G\\'abor Somlai (E\\"otv\\"os Lor\\' and University and R\\'enyi I
nstitute\, Hungary)
DTSTART:20240522T130000Z
DTEND:20240522T132500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/16
DESCRIPTION:Title: New method for old results of R\\'edei\, Lov\
\'asz and Schrijver\nby G\\'abor Somlai (E\\"otv\\"os Lor\\' and Unive
rsity and R\\'enyi Institute\, Hungary) as part of Combinatorial and addit
ive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 i
n the CUNY Graduate Center.\n\nAbstract\nR\\'edei proved that a set $S$ of
cardinality $p$ in $\\mathbb{F}_p^2$ determines at least $\\frac{p+3}{2}$
directions or $S$ is a line. \nWe managed find a short proof for R\\'ede
i's result avoiding the theory of lacunary polynomials by proving the foll
owing statement. Let $f $ be a polynomial over the finite field $\\mathbb{
F}_p$. Consider the elements of the range as integers in $\\{0\,1\, \\ldot
s\, p-1 \\}$. Assume that $\\sum_{x \\in \\mathbb{F}_p}f(x)=p$. Then eithe
r $f=1$ or $deg(f) \\ge \\frac{p-1}{2}$.\nThe uniqueness (up to affine tra
nsformations) of the sets of size $p$\nin $\\mathbb{F}_p^2$ was proved by
Lov\\'asz and Schrijver. The same result follows from the almost uniquenes
s of the polynomials of degree $\\frac{p-1}{2}$ of range sum $p$.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Konyagin (Steklov Institute of Mathematics\, Russia)
DTSTART:20240522T133000Z
DTEND:20240522T135500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/17
DESCRIPTION:Title: On distinct angles in the plane\nby Serge
i Konyagin (Steklov Institute of Mathematics\, Russia) as part of Combinat
orial and additive number theory (CANT 2024)\n\nLecture held in Room 4102
and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nThe talk is based
on our joint paper with Jonathan Passant and Misha Rudnev. \nWe prove that
if $N$ points lie in convex position in the plane\, then they determine\n
$\\gg N^{1+3/23+o(1)}$ distinct angles\, provided no $N-1$ points lie on a
common circle.\nThis is the first super--linear bound on the distinct ang
le problem that has received\nrecent attention.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krystian Gajdzica (Jagiellonian University\, Krak\\' ow\, Poland)
DTSTART:20240522T140000Z
DTEND:20240522T142500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/18
DESCRIPTION:Title: A combinatorial approach to the Bessenrodt-On
o type inequalities\nby Krystian Gajdzica (Jagiellonian University\, K
rak\\' ow\, Poland) as part of Combinatorial and additive number theory (C
ANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate
Center.\n\nAbstract\nIn 2016\, Bessenrodt and Ono showed that the partitio
n function satisfies the inequality of the form\n $$p(a)p(b)>p(a+b)$$\n
for all $a\,b\\geqslant2$ with $a+b>9$. Their proof is based on the asympt
otic estimates of $p(n)$ due to Lehmer. Since then\, a lot of similar phen
omena have been discovered for various variations of the partition functio
n. \n\nWe discuss the analogue of the Bessenrodt-Ono inequality for the so
-called $A$-partition function $p_A(n)$\, which enumerates those partition
s of $n$ whose parts belong to a fixed set $A\\subset\\mathbb{N}$. Since t
here is no known asymptotic formula for $p_A(n)$ in general\, we can not d
eal with the problem using any estimates of $p_A(n)$. Therefore\, we prese
nt a combinatorial approach to the issue by constructing an appropriate in
jection between some sets of partitions.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rauan Kaldybayev (Williams College)
DTSTART:20240522T150000Z
DTEND:20240522T152500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/19
DESCRIPTION:Title: Limiting behavior in missing sums of sumsets<
/a>\nby Rauan Kaldybayev (Williams College) as part of Combinatorial and a
dditive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 92
07 in the CUNY Graduate Center.\n\nAbstract\nWe study $|A + A|$ as a rando
m variable\, where $A \\subseteq \\{0\, \\dots\, N\\}$ is a random subset
such\n that each\n $0 \\le n \\le N$ is included with probability $0 < p
< 1$\, and where $A + A$ is the set of sums $a + b$ for $a\,b$\n in $A$.\n
Lazarev\, Miller\, and O'Bryant studied the distribution of $2N + 1 - |A
+ A|$\, the number of summands not\n represented in\n $A + A$ when $p =
1/2$. A recent paper by Chu\, King\, Luntzlara\, Martinez\, Miller\, Shao\
, Sun\, and Xu generalizes\n this to\n all $p\\in (0\,1)$\, calculating t
he first and second moments of the number of missing summands and establis
hing\n exponential upper and lower bounds on the probability of missing e
xactly $n$ summands\, mostly working in the\n limit of\n large $N$. We pr
ovide exponential bounds on the probability of missing at least $n$ summan
ds\, find another\n expression\n for the second moment of the number of m
issing summands\, extract its leading-order behavior in the limit of\n sma
ll $p$\,\n and show that the variance grows asymptotically slower than th
e mean\, proving that for small $p$\, the number of\n missing\n summands
is very likely to be near its expected value.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian T\\'afula (Universit\\'e de Montr\\'eal\, Canada)
DTSTART:20240522T160000Z
DTEND:20240522T162500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/20
DESCRIPTION:Title: Representation functions with prescribed rate
s of growth\nby Christian T\\'afula (Universit\\'e de Montr\\'eal\, Ca
nada) as part of Combinatorial and additive number theory (CANT 2024)\n\nL
ecture held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbs
tract\nLet $h\\geq 2$\, and $b_1$\, $\\ldots$\, $b_h$ be positive integers
with $\\gcd = 1$. For a set $A\\subseteq \\mathbb{N}$\, denote by $r_A(n)
$ the number of solutions to the equation\n \\[ b_1 k_1 + ... + b_h k_h =
n \\]\n with $k_1$\, $\\ldots$\, $k_h\\in A$. For which functions $F$ can
we find $A$ such that $r_A(n) \\sim F(n)$? Or $r_A(n)\\asymp F(n)$? In the
asymptotic case\, we show that for every $F$ of regular variation satisfy
ing\n \\[ \\frac{F(x)}{\\log x} \\xrightarrow{x\\to\\infty} \\infty\, \\qu
ad\\text{and}\\quad F(x) \\leq (1 + o(1)) \\dfrac{x^{h-1}}{(h-1)!b_1...b_h
}\, \\]\n there is $A$ such that $r_A(n) \\sim F(n)$. In the order of magn
itude case\, there is $A$ with $r_A(n)\\asymp F(x)$ for every $F$ non-decr
easing such that $F(2x)\\ll F(x)$ in the range $\\log x \\ll F(x) \\ll x^{
h-1}$. This extends earlier work of Erdős–Tetali and Vu\, and addresses
a question raised by Nathanson on which functions can be the $r_A$ of som
e $A$.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Lebowitz-Lockard (University of Texas\, Tyler\, TX)
DTSTART:20240522T173000Z
DTEND:20240522T175500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/21
DESCRIPTION:Title: On the smallest parts of partitions into dist
inct parts\nby Noah Lebowitz-Lockard (University of Texas\, Tyler\, TX
) as part of Combinatorial and additive number theory (CANT 2024)\n\nLectu
re held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstrac
t\nFor a given integer $n$\, let $D(n)$ be the set of partitions of $n$ in
to distinct parts. Create a sum as follows. For each partition $\\lambda$
in $D(n)$\, add the smallest element of $\\lambda$ if it is even and subtr
act it if it is odd. A classic theorem of Uchimura states that this quanti
ty is equal to the number of divisors of $n$. We generalize this result to
the sum of the $k$th smallest elements of partitions for a fixed value of
$k$. We also consider some further generalizations\, as well as variants
for the smallest number not in a given partition. \\\\\nJoint work with Ra
jat Gupta and Joseph Vandehey.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Donley (Queensborough Community College (CUNY))
DTSTART:20240522T180000Z
DTEND:20240522T182500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/22
DESCRIPTION:Title: A combinatorial introduction to adinkras\
nby Robert Donley (Queensborough Community College (CUNY)) as part of Comb
inatorial and additive number theory (CANT 2024)\n\nLecture held in Room 4
102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nIn 2005\, Faux
and Gates defined the adinkra\, a graphical device for describing particl
e exchanges in supersymmetry. Independent of the physical applications\,
the adinkra resides at a nexus of various mathematical concepts and proble
ms. In this talk\, we give an introduction to adinkras from the point of
view of matching problems in combinatorics. \n\nJoint work with S. James
Gates\, Jr.\, Tristan H{\\"u}bsch\, and Rishi Nath.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Flores (Purdue University)
DTSTART:20240522T183000Z
DTEND:20240522T185500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/23
DESCRIPTION:Title: A circle method approach to K-multimagic squa
res\nby Daniel Flores (Purdue University) as part of Combinatorial and
additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room
9207 in the CUNY Graduate Center.\n\nAbstract\nWe investigate $K$-multimag
ic squares of order $N$\, which are $N \\times N$ magic squares with remai
n magic after raising each element to the $k$th power for all $2 \\le k \\
le K$. Given $K \\ge 2$\, we consider the problem of establishing the smal
lest integer $N(K)$ for which there exists nontrivial $K$-multimagic squar
es of order $N(K)$. Previous results on multimagic squares show that $N(K)
\\le (4K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle
method and establish the bound \n\\[N(K) \\le 2K(K+1)+1.\\] \n\nVia an ar
gument of Granville's we additionally deduce the existence of infinitely m
any \\emph{non-trivial} prime valued $K$-multimagic squares of order $2K(K
+1)+1$.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Fel (Technion -- Israel Institute of Technology\, Israel)
DTSTART:20240522T190000Z
DTEND:20240522T192500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/24
DESCRIPTION:Title: Ratio between a sum of generators and rationa
l powers of their product\nby Leonid Fel (Technion -- Israel Institute
of Technology\, Israel) as part of Combinatorial and additive number theo
ry (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Grad
uate Center.\n\nAbstract\nWe study a ratio $R_m(k)\\!= I_1/\\sqrt[k]{I_m}$
between a sum $I_1\\!=\\!\n\\sum_{j=1}^md_j$ of generators \nand rational
powers $\\sqrt[k]{I_m}$ of their product $I_m=\\prod_{j=1}^md_j$ in numer
ical \nsemigroups $\\langle d_1\,\\ldots\,d_m\\rangle$. We find its upper
${\\sf R}_m^+(k)$ and lower ${\\sf R}_m^-(k)$ bounds \n in the range $1\\l
e k\\le m$. We prove that $R_m(k)$ has a universal upper bound \nif and o
nly if $m\\ge 2k-1$.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksei Volostnov (Moscow Institute of Physics and Technology\, Ru
ssia)
DTSTART:20240522T193000Z
DTEND:20240522T195500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/25
DESCRIPTION:Title: On the additive energy of roots\nby Aleks
ei Volostnov (Moscow Institute of Physics and Technology\, Russia) as part
of Combinatorial and additive number theory (CANT 2024)\n\nLecture held i
n Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $p
$ be a prime number\, $f\\in \\mathbb F_p[x]$ be a polynomial of small deg
ree and a set $A\\subset \\mathbb F_p$ have sufficiently small cardinality
in terms of $p$. \nWe study the number of solutions to the equation (in $
\\mathbb F_p$)\n\\[\nx_1+x_2 = x_3+x_4\,\\quad f(x_1)\,f(x_2)\,f(x_3)\,f(x
_4)\\in A\,\n\\]\nprovided that $A$ has small doubling. Namely\, we improv
e the upper bound \nfrom recent work by B. Kerr\, I. D. Shkredov\, I. E. S
hparlinski and A. Zaharescu.\n\nMoreover\, we address questions of cardina
lities $|A+A|$ vs $|f(A)+f(A)|$. \nIn particular\, we prove that \n\\[\n
\\max(|A+A|\,|A^3+A^3|)\\gg |A|^{16/15} \n \\] \n \\[\n \\max
(|A+A|\,|A^4 +A^4|)\\gg |A|^{25/24} \n\\]\n \\[\n \\max(|A+A|\,|A^5
+A^5|)\\gg |A|^{25/24}. \n\\]\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20240522T200000Z
DTEND:20240522T202500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/26
DESCRIPTION:Title: Matchable numbers\nby Carl Pomerance (Dar
tmouth College) as part of Combinatorial and additive number theory (CANT
2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Cent
er.\n\nAbstract\nFor a natural number $n$ let $D(n)$ denote the set of pos
itive divisors of $n$ and\nlet $\\tau(n)=\\#D(n)$. Say $n$ is {\\it match
able} if there is a bijection from\n$D(n)$ to $\\{1\,2\,\\dots\,\\tau(n)\\
}$ with corresponding numbers relatively prime.\nFor example\, each number
up to 7 is matchable\, but 8 is not.\nThis definition was made by Santos
on MathOverflow in 2022\; he\nasks if there are more matchable numbers \nt
han not. We prove this by showing the set of matchable numbers has an asy
mptotic density\ngiven by $\\prod_{p\\\,{\\rm prime}}(1-1/p^p)=.72199\\dot
s$. \\\\\nThis is joint work with Nathan McNew.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Miller (Williams College)
DTSTART:20240522T203000Z
DTEND:20240522T205500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/27
DESCRIPTION:Title: The theory of normalization constants and Zec
kendorf decompositions\nby Steve Miller (Williams College) as part of
Combinatorial and additive number theory (CANT 2024)\n\nLecture held in Ro
om 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nIf we defi
ne the Fibonacci numbers to start 1\, 2\, 3\, 5 and so on\, we have a wond
erful property: Every positive integer has a unique representation as a su
m of non-adjacent terms. Called the Zeckendorf decomposition\, we can prov
e many results about the summands\, from the number in a typical decomposi
tion converging to a Gaussian to the probabilities of gaps converging to a
geometric decay. Many of these proofs are straightforward but tedious exe
rcises in algebra. We present a new approach\, which so far has just been
applied to the distribution of gaps\, but hopefully can work for related
problems\, which bypasses these calculations through the theory of normali
zation constants.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Sellers (University of Minnesota Duluth)
DTSTART:20240522T143000Z
DTEND:20240522T145500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/28
DESCRIPTION:Title: Elementary proofs of congruences for POND and
PEND partitions\nby James Sellers (University of Minnesota Duluth) as
part of Combinatorial and additive number theory (CANT 2024)\n\nLecture h
eld in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nR
ecently\, Ballantine and Welch considered two classes of integer partition
s which they labeled POND and PEND partitions. These are integer partition
s wherein the odd parts (respectively\, the even parts) cannot be distinct
. In recent work\, I studied these two types of partitions from an arithme
tic perspective and proved infinite families of mod 3 congruences satisfie
d by the two corresponding enumerating functions. I will talk about the ge
nerating functions for these enumerating functions\, and I will also highl
ight the (induction) proofs that I utilized.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jin-Hui Fang (Nanjing Normal University\, China)
DTSTART:20240523T130000Z
DTEND:20240523T132500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/29
DESCRIPTION:Title: On Cilleruelo-Nathanson's method in Sidon set
s\nby Jin-Hui Fang (Nanjing Normal University\, China) as part of Comb
inatorial and additive number theory (CANT 2024)\n\nLecture held in Room 4
102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nFor nonnegativ
e integers $h\,g$ with $h\\ge 2$\, a set $\\mathcal{A}$ of nonnegative int
egers is defined as a $B_h[g]$ sequence if\, for every nonnegative integer
$n$\, the number of representations of $n$ with the form $n=a_1+a_2+\\cdo
ts+a_h$ is no larger than $g$\, where $a_1\\le \\cdots \\le a_h$ and $a_i\
\in \\mathcal{A}$ for $i=1\,2\,\\cdots\,h$. Let $\\mathbb{Z}$ be the set o
f integers and $\\mathbb{N}$ be the set of positive integers. In 2013\, b
y introducing the method of \\emph{Inserting Zeros Transformation}\, Cille
ruelo and Nathanson obtained the following nice result: let $f:\\mathbb{Z}
\\rightarrow \\mathbb{N}\\bigcup \\{0\,\\infty\\}$ be any function such th
at $\\liminf_{|n|\\rightarrow \\infty} f(n)\\ge g$ and let $\\mathcal{B}$
be any $B_h[g]$ sequence. Then\, for any decreasing function $\\epsilon(x)
\\rightarrow 0$ as $x\\rightarrow \\infty$\, there exists a sequence $\\ma
thcal{A}$ of integers such that $r_{\\mathcal{A}\,h}(n)=f(n)$ for all $n\\
in \\mathbb{Z}$ and $\\mathcal{A}(x)\\gg B(x\\epsilon(x))$. In 2022\, Nath
anson further considered Sidon sets for linear forms. Recently\, we apply
the Inserting Zeros Transformation into Sidon sets for linear forms and ge
neralize the above result related to the inverse problem of representation
functions.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:N. Hegyv\\'ari (E\\"otv\\"os Lor\\' and University and R\\'enyi In
stitute\, Hungary)
DTSTART:20240523T133000Z
DTEND:20240523T135500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/30
DESCRIPTION:Title: On the structures of sets in $\\mathbb{N}^k$
having thin subset sums\nby N. Hegyv\\'ari (E\\"otv\\"os Lor\\' and Un
iversity and R\\'enyi Institute\, Hungary) as part of Combinatorial and ad
ditive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 920
7 in the CUNY Graduate Center.\n\nAbstract\nFor any $X\\subseteq \\mathbb{
N}^k$ let\n\\[\nFS(X):=\\{\\sum_{i=1}^\\infty\\varepsilon_ix_i: \\ x_i\\in
X\, \\ \\varepsilon_i \\in \\{0\,1\\}\, \\ \\sum_{i=1}^\\infty\\varepsilo
n_i<\\infty\\}\n\\]\nErdős called a sequence $A\\subseteq \\mathbb{N}$ co
mplete if every sufficiently large number belongs to $FS(A)$. \nIn a high
er dimension too\, the necessary condition that the subset sums of a subse
t $X\\subseteq \\mathbb{N}^k$ represent all far points of $\\mathbb{N}^k$
should be the condition $X(N)>k\\log_2N+t_X$ for some $t_X$\, i.e. $X$ is
complete respect to the region $R=\\{x=(x_1\,x_2\,\\dots\,x_k):x_i\\geq r_
i\\}$\, $r_i\\in \\mathbb{N}$\, $i=1\,2\,\\dots\, k$.\n\n\n We say that $A
$ is weakly thin if $\\limsup_{n\\to \\infty }\\frac{\\log a_n}{\\log n}=\
\infty$\, or equivalently $A(n):= \\sum_{a_i\\leq n}1=n^{g(n)}$\, where $A
(n)$ is the counting function of $A$ and $\\liminf_{n\\to \\infty }g(n)=0$
. \nA set $B\\subseteq \\mathbb{N}$ is said to be thick if it is not weakl
y thin.\nLet $X\\subseteq \\mathbb{N}^k$.\n$X$ is said to be thin complete
set respect to $R$ if $X(N)>k\\log_2R(N)+t_X$ for some $t_X$ and $FS(X)\
\supseteq R$. \n\nWe prove that if $R=z+\\mathbb{N}^k$ and $A$ is complete
with respect to $R$\, then all projections of $A$ onto for all axis $f_i$
are thick.\nWe also determine the regions for which thin complete sets ex
ist. We also invetigate the structure of $FS(Y)$\, where $Y=\\{a_m\\}_{m\\
in \n}\\times\\{b_k\\}_{k\\in \n}$ and $\\{a_m\\}_{m\\in \n}$ is a 'dense'
sequence.\n\n\n\nThis is a joint work with Máté Pálfy and Erfei Yue.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sukumar Das Adhikari (Ramakrishna Mission Vivekananda Educational
and Research Institute (RKMVERI)\, India)
DTSTART:20240523T140000Z
DTEND:20240523T142500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/31
DESCRIPTION:Title: Some elementary algebraic and combinatorial m
ethods in the study of zero-sum theorems\nby Sukumar Das Adhikari (Ram
akrishna Mission Vivekananda Educational and Research Institute (RKMVERI)\
, India) as part of Combinatorial and additive number theory (CANT 2024)\n
\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\n
Abstract\nOriginating from a beautiful theorem of Erdos-Ginzberg-Ziv about
sixty years ago and some other \nquestions asked around the same time\, t
he area of zero-sum theorems has many interesting results \nand several un
answered questions.\n\nSeveral authors have introduced interesting element
ary algebraic techniques to deal with these problems.\nWe describe some ex
periments with these elementary algebraic methods and some combinatorial o
nes\, \nin a weighted generalization in the area of Zero-sum Combinatoric
s.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Leonetti (Università degli Studi dell'Insubria\, Italy)
DTSTART:20240523T143000Z
DTEND:20240523T145500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/32
DESCRIPTION:Title: Most numbers are not normal\nby Paolo Leo
netti (Università degli Studi dell'Insubria\, Italy) as part of Combinato
rial and additive number theory (CANT 2024)\n\nLecture held in Room 4102 a
nd Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $S$ be the set
of real numbers $x \\in (0\,1]$ with the following property of being \\tex
tquotedblleft strongly not normal": \nFor all integers $b\\ge 2$ and $k\\g
e 1$\, the sequence of vectors made by the frequencies of all possible str
ings of length $k$ in the $b$-adic representation of $x$ has a maximal sub
set of accumulation points\, and each of them is the limit of a subsequenc
e with an index set of nonzero asymptotic density. \n\\vskip 0\,05cm\nWe s
how that $S$ is a co-meager subset of $(0\,1]$\, hence topologically large
. Analogues are given in the context of regular matrices.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Dolores Cuenca (Pusan National University\, Korea)
DTSTART:20240523T150000Z
DTEND:20240523T152500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/33
DESCRIPTION:Title: Zeta values as an algebra over an operad\
nby Eric Dolores Cuenca (Pusan National University\, Korea) as part of Com
binatorial and additive number theory (CANT 2024)\n\nLecture held in Room
4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nDenote the op
erad of finite posets by FP. In number theory\, the field of rational zet
a series studies series of the form $\\sum_{i=1}^\\infty a_i (\\zeta(i+1)-
1)\, a_i\\in\\mathbb{Q}\\\,\\forall i\\in\\mathbb{N}$\, where $\\zeta(k)$
is the Riemann zeta function $\\zeta(k)=\\sum_{n=1}^\\infty\\frac{1}{n^k}$
. By studying zeta values as algebras over the operad of posets\,\nwe show
the following identity\, for $a>1\, a\\in\\mathbb{N}:$\n\n$$\\sum_{n=i}^\
\infty (-1)^{n+1}{n\\choose i}\\zeta(n+1\,a)=(-1)^{i+1}\\zeta(i+1\,a+1)\,$
$\nhere\, $\\zeta(k\,a)=\\sum_{n=0}^\\infty\\frac{1}{(n+a)^k}$ is the Hurw
itz zeta function. \n\nOn January 2023 we put the left side of the identit
y on several private software\, but none of them produced any output. We p
resented our work in the Wolfram Technology Conference 2023\, where their
team kindly verified that the left side of the identity is equal to the ri
ght side of the identity. \\\\\nJoint work with Jose Mendoza-Cortes\, Mich
igan State University\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qitong (George) Luan (University of California\, Los Angeles)
DTSTART:20240523T153000Z
DTEND:20240523T155500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/34
DESCRIPTION:Title: On a pair of diophantine equations\nby Qi
tong (George) Luan (University of California\, Los Angeles) as part of Com
binatorial and additive number theory (CANT 2024)\n\nLecture held in Room
4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nFor relativel
y prime natural numbers $a$ and $b$\, we study the two equations $ax+by =
(a-1)(b-1)/2$ and $ax+by+1=\n(a-1)(b-1)/2$\, which arise from the study of
cyclotomic polynomials. Previous work showed that exactly one equation ha
s\na nonnegative integer solution\, and the solution is unique. Our first
result gives criteria to determine which equation\nis used for a given pai
r $(a\,b)$. We then use the criteria to study the sequence of equations us
ed by the pair\n$(a_n/\\gcd{(a_n\, a_{n+1})}\, a_{n+1}/\\gcd{(a_n\, a_{n+1
})})$ from several special sequences $(a_n)_{n\\geq 1}$\, such as\narithme
tic progressions\, geometric progressions and sequences satisfying Fibonac
ci-type recurrences. Furthermore\, for\neach positive $k$\, we construct a
sequence $(a_n)_{n}$ whose consecutive terms use the two equations altern
atively in\ngroups of $k$. Lastly\, we investigate the periodicity of the
sequence of equations used by the pair $(k/\\gcd{(k\, n)}\,\nn/\\gcd{(k\,
n)})$ as $n$ increases.\\\\\nJoint work with Sujith Uthsara Kalansuriya A
rachchi\, \nH\\`ung Vi\\d{\\^e}t Chu\, Jiasen Liu\, Rukshan Marasinghe\, a
nd Steven J. Miller.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20240523T160000Z
DTEND:20240523T162500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/35
DESCRIPTION:Title: Local distance-resistance functions equivalen
t to global symmetries in electric circuit families\nby Russell Jay He
ndel (Towson University) as part of Combinatorial and additive number theo
ry (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Grad
uate Center.\n\nAbstract\nIn a recent paper Hendel explored the computatio
nal attributes of an algorithm introduced by Barrett\, Evans\, and Francis
\, which\, among other things\, studied distance resistance in a family of
circuits whose underlying graphs consisted of $n$ rows of upright equilat
eral triangles ($n$-grids). Two important conjectures supported by numeric
al evidence were presented: one related to the asymptotic behavior of iter
ated use of the algorithm on an initial $n$ grid as $n$ goes to infinity.
The second conjecture showed that as $n$ grows large certain limiting rat
ios emerge among specified edges in the circuits resulting from a large nu
mber of repeated applications of the algorithm to an initial $n$-grid. The
purpose of this paper is to provide insight into these asymptotic or lim
iting edge ratios. After introducing the algorithm and reviewing the origi
nal conjectures\, the main part of this paper studies a family of $n$-gri
ds whose edge labels are determined using these limiting edge-ratios fun
ctions. The main result proven is that these $n$-grids as well as the
graphs derived from repeated application of the algorithm possess verti
cal and rotational symmetries and also continue to satisfy the relationshi
ps captured by the limiting edge-ratio functions. In other words\, the lim
iting edge-ratio relationships are local algebraic relationships mirrorin
g the global vertical and rotational symmetries possessed by the underlyin
g graph. Additionally\, because row-reduction is local (in contrast to the
combinatoric Laplacian which is global) the paper is able to introduce a
mechanical verification method of proof for assertions about effective re
sistance identities.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY)\, The Graduate Cen
ter (CUNY))
DTSTART:20240523T173000Z
DTEND:20240523T175500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/36
DESCRIPTION:Title: Greedy $B_h$-sets\nby Kevin O'Bryant (Col
lege of Staten Island (CUNY)\, The Graduate Center (CUNY)) as part of Comb
inatorial and additive number theory (CANT 2024)\n\nLecture held in Room 4
102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nA set $X$ of i
ntegers is a $B_h$-set if every solution to $\na_1+\\cdots+a_h=b_1+\\cdots
+b_h$ with $a_i\,b_i\\in X$ has\n$\\{a_1\,\\ldots\,a_h\\}=\\{b_1\,\\dots\
,b_h\\}$ (as multisets). The main\nproblem is to give inequalities connect
ing the cardinality and\ndiameter of $B_h$-sets\, and one obvious way to b
uild thick $B_h$-sets\nis to be greedy. In this talk we survey old and new
results on the\ngreedy $B_h$-sets. The highlight of the new results is a
nontrivial\nupper bound on the $k$-th element of the greedy $B_h$-set\, pr
ovided\nthat $h$ is sufficiently large.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (Stanford University)
DTSTART:20240523T180000Z
DTEND:20240523T182500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/37
DESCRIPTION:Title: Goldbach beyond the square-root barrier\n
by Jared Duker Lichtman (Stanford University) as part of Combinatorial and
additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room
9207 in the CUNY Graduate Center.\n\nAbstract\nWe show the primes have lev
el of distribution 66/107 using triply well-factorable weights\, and exten
d this level to 5/8 assuming Selberg's eigenvalue conjecture. This improve
s on the prior world record level of 3/5 by Maynard. As a result\, we obta
in new upper bounds for Goldbach representations of even numbers. This is
the first use of a level of distribution beyond the 'square-root barrier'
for the Goldbach problem\, and leads to the greatest improvement on the pr
oblem since Bombieri-Davenport from 1966.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor D. Wooley (Purdue University)
DTSTART:20240523T183000Z
DTEND:20240523T185500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/38
DESCRIPTION:Title: Unrepresentation theory and sums of powers\nby Trevor D. Wooley (Purdue University) as part of Combinatorial and ad
ditive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 920
7 in the CUNY Graduate Center.\n\nAbstract\nWe report on recent and on-goi
ng work joint with J\\" org Br\\" udern concerning problems involving the
representation of integer sequences by sums of powers. Our new tool is an
upper bound for moments of smooth Weyl sums restricted to major arcs. This
permits progress to be made on Waring's problem and other problems involv
ing mixed sums of powers and primes. We will focus on recent progress conc
erning unrepresentation theory (bounds for exceptional sets).\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brad Isaacson (NYC College of Technology (CUNY))
DTSTART:20240523T190000Z
DTEND:20240523T192500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/39
DESCRIPTION:Title: On a reciprocity formula for generalized Dede
kind-Rademacher sums attached to three Dirichlet characters\nby Brad I
saacson (NYC College of Technology (CUNY)) as part of Combinatorial and ad
ditive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 920
7 in the CUNY Graduate Center.\n\nAbstract\nWe define a three character an
alogue of the generalized Dedekind-Rademacher sum introduced by Hall\, Wil
son\, and Zagier\, and state its reciprocity formula\, which contains all
of the reciprocity formulas in the literature for generalized Dedekind-Rad
emacher sums attached (and not attached) to Dirichlet characters as specia
l cases. We also review some of the generalized Dedekind-Rademacher sums
in the literature to motivate our results.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Firdavs Rakhmonov (University of Rochester)
DTSTART:20240523T200000Z
DTEND:20240523T202500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/41
DESCRIPTION:Title: The quotient set of the quadratic distance se
t over finite fields\nby Firdavs Rakhmonov (University of Rochester) a
s part of Combinatorial and additive number theory (CANT 2024)\n\nLecture
held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\n
Let $\\mathbb F_q^d$ be the $d$-dimensional vector space over the finite f
ield $\\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\\mathbb
F_q$ and $E\\subset \\mathbb F_q^d$\, we define $W(r)$ as the number of q
uadruples $(x\,y\,z\,w)\\in E^4$ such that $\nQ(x-y)/Q(z-w)=r\,$ where $Q$
is a non-degenerate quadratic form in $d$ variables over $\\mathbb F_q.$\
nWhen $Q(\\alpha)=\\sum_{i=1}^d \\alpha_i^2$ with $\\alpha=(\\alpha_1\, \\
ldots\, \\alpha_d)\\in \\mathbb F_q^d\,$ \nPham (2022) recently used the m
achinery of group actions and proved that if $E\\subset \\mathbb F_q^2$ w
ith $q\\equiv 3 \\pmod{4}$ and $|E|\\ge C q$\, then we have $W(r)\\ge c |E
|^4/q$ for any non-zero square number $r \\in \\mathbb F_q\,$ where $C$ i
s a sufficiently large constant\, $ c$ is some number between $0$ and $1\,
$ and $|E|$ denotes the cardinality of the set $E.$\n%In this talk\, \nI'l
l discuss the improvement and extension of Pham's result in two dimensions
to arbitrary dimensions with general non-degenerate quadratic distances.
As a corollary\, we also generalize the sharp results on the Falconer typ
e problem for the quotient set of distance set due to Iosevich-Koh-Parshal
l. Furthermore\, we provide improved constants for the size conditions of
the underlying sets.\nJoint work with Alex Iosevich and Doowon Koh.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART:20240523T203000Z
DTEND:20240523T205500Z
DTSTAMP:20260404T094149Z
UID:NumberTheoryAndCombinatorics/42
DESCRIPTION:Title: Can you reconstruct a set from its subset sum
s?\nby Noah Kravitz (Princeton University) as part of Combinatorial an
d additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room
9207 in the CUNY Graduate Center.\n\nAbstract\nFor a finite subset $A$ of
an abelian group\, let $\\text{FS}(A)$ denote the multiset of its $2^{|A|
}$ subset sums. Can you reconstruct $A$ from $\\text{FS}(A)?$ The answer i
n general is ``no'' (for instance\, $\\text{FS}(\\{1\,3\,-4\\})=\\text{FS}
(\\{-1\,-3\,4\\})$)\, but in many cases\, such as when the ambient group h
as no $2$-torsion\, we can obtain a combinatorial description of the fiber
s of $\\text{FS}$. \n\nJoint work with Federico Glaudo.\n
LOCATION:https://stable.researchseminars.org/talk/NumberTheoryAndCombinato
rics/42/
END:VEVENT
END:VCALENDAR