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BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (CUNY)
DTSTART:20260713T130000Z
DTEND:20260713T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/1
DESCRIPTION:Title: The Sidon error term\nby Kevin O'Bryant (CUNY) as part of Combinatori
al and additive number theory seminar (CANT 2026)\n\nLecture held in Scien
ce Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nA Sidon se
t is a set ${\\mathcal A}$ of integers that has no nontrivial solutions to
$a+b=c+d$. It has been known since 1941 (Erd\\H{o}s and Tur\\'an) that if
${\\mathcal A}$ is a finite Sidon set\, then $\\text{diam}({\\mathcal A})
\\ge k^2 - 2k^{3/2} + O(k)$\, and since 1939 (Singer) that the $k^2$ term
cannot be improved. Only in the last 5 years has the error term $-2k^{3/2
}$ been sharpened (Balogh\, F\\"uredi\, and Roy\, then O'Bryant\, then Car
ter\, Hunter\, O'Bryant). In this talk\, I will relay the latest improveme
nts and applications\, and the use of AI (AlphaEvolve) in their discovery.
Joint work with D.~Carter\, B.~Georgiev\, Z.~Hunter\, J.~G.~Serrano\, T.
~Tao\, and A.~Zs.~Wagner.\n
LOCATION:https://researchseminars.org/talk/CANT2026/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aradhya Goel (Indian Institute of Technology Kanpur)
DTSTART:20260713T133000Z
DTEND:20260713T135500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/2
DESCRIPTION:Title: Sophie Germain primes and the totient of Fibonacci numbers\nby Aradhy
a Goel (Indian Institute of Technology Kanpur) as part of Combinatorial an
d additive number theory seminar (CANT 2026)\n\nLecture held in Science Ce
nter in the CUNY Graduate Center (4th floor).\n\nAbstract\nWe study the se
t $S(q)$ of residue classes $r$ modulo the Pisano period $\\pi(q)$ for whi
ch $q \\mid \\varphi(F_m)$ for every $m \\equiv r \\pmod{\\pi(q)}$. We pro
ve that if $q$ is a Sophie Germain prime and $z(2q+1) \\mid \\pi(q)$\, whe
re $z$ denotes the rank of apparition\, then $S(q)$ is a nonempty arithmet
ic progression\; for $q > 5$\, its cardinality is odd and $q \\equiv 8 \\p
mod{15}$. Conversely\, if a prime $p \\equiv 1 \\pmod{q}$ has $z(p) \\mid
\\pi(q)$\, then necessarily $p = 2q+1$\, so $q$ is Sophie Germain. \nWe co
njecture that $S(q) \\neq \\emptyset$ forces the existence of such a prime
$p$\; this is verified for all $q \\leq 50{\,}000$. Assuming the divisibi
lity $z(2q+1) \\mid \\pi(q)$ holds for infinitely many Sophie Germain prim
es (verified for approximately $23.9\\%$ of the $669$ Sophie Germain prime
s $q \\leq 50{\,}000$)\, the Sophie Germain conjecture implies the existen
ce of infinitely many primes $q \\equiv 8 \\pmod{15}$ with $(2q+1) \\mid F
_{\\pi(q)}$ -- a purely Fibonacci-theoretic condition. \nThese results gen
eralize to arbitrary Lucas sequences $U_n(P\,Q)$ with non-square discrimin
ant.\n
LOCATION:https://researchseminars.org/talk/CANT2026/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Pal Pach (Renyi Institute)
DTSTART:20260713T140000Z
DTEND:20260713T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/3
DESCRIPTION:Title: On the density of Kravitz sets\nby Peter Pal Pach (Renyi Institute) a
s part of Combinatorial and additive number theory seminar (CANT 2026)\n\n
Lecture held in Science Center in the CUNY Graduate Center (4th floor).\n\
nAbstract\nWe show that for a subset $A$ of the cyclic group of prime orde
r $p>3$\, if the sumset $A+A-2A=\\{a_1+a_2-2a_3:\\ a_1\,a_2\,a_3 \\in A\\
}$ is not the whole group\, then $|A|\\le \\frac27\\\,p$. \nBesides comb
inatorial arguments\, we utilize a general technique involving linear pro
gramming\, which may find further applications in additive combinatorics
in the future. \n Joint work with Vsevolod Lev\, Mate Matolcsi\, Daniel V
arga.\n
LOCATION:https://researchseminars.org/talk/CANT2026/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jose Ramon Madrid Padilla (Virginia Tech)
DTSTART:20260713T143000Z
DTEND:20260713T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/4
DESCRIPTION:Title: Convolution inequalities and applications\nby Jose Ramon Madrid Padil
la (Virginia Tech) as part of Combinatorial and additive number theory sem
inar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Ce
nter (4th floor).\n\nAbstract\nIn this talk\, we will discuss a collection
of optimal convolution inequalities for real-valued functions on the hype
rcube\, motivated by combinatorial applications. In particular\, as a cons
equence we obtain sharp bounds for sumsets and additive energies of subset
s of the hypercube.\n
LOCATION:https://researchseminars.org/talk/CANT2026/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Grynkiewicz (Memphis University)
DTSTART:20260713T160000Z
DTEND:20260713T162500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/7
DESCRIPTION:Title: On factorizations of zero-sum sequences over abelian torsion groups\
nby David Grynkiewicz (Memphis University) as part of Combinatorial and ad
ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center
in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $G$ be an addit
ive abelian torsion group and let $G_0\\subseteq G$ be a subset. A zero-su
m sequence over $G_0$ is an unordered string of terms from $G_0$ (repetiti
on of terms allowed) such that the sum of terms is $0$. In the last few de
cades\, the connection between factorizations of zero-sum sequences and fa
ctorization of elements in rings of integers has been made more precise an
d extended into much more general algebraic settings. The extent to which
factorization are wild or well-behaved is often measured by the finiteness
and size of various arithmetic factorization invariants. Some of the most
common include the catenary degree $\\mathsf c(G_0)$\, the set of success
ive distances $\\Delta(G_0)$\, and the elastacities $\\rho_k(G_0)$. We beg
in by introducing what these invariants are in purely combinatorial terms
and explain how they measure constraint of factorization in algebraic sett
ings. \n\nIn the past\, there has been much focus on finite groups\, and m
ore recently\, on subsets of finitely generated groups. However\, very lit
tle was known in the case of non-finitely generated abelian groups. In par
t\, this is because common invariants used to study factorization\, such a
s the Davenport Constant\, are no longer guaranteed to be finite. In order
to better understand factorization in the setting of infinite abelian tor
sion groups\, we introduce a new technique measuring the size of a sequenc
e not by the number of its terms but rather by its cross number\, $\\sum_{
i=1}^{\\ell} \\frac{1}{\\text{\\rm ord} (g_i)}$\, where the $g_i\\in G_0\\
subseteq G$ are the terms in the sequence. The use of cross numbers allows
us to define three constants\, $\\mathsf K(G_0)$\, $\\mathsf k(G_0)$ and
$\\mathsf K_{\\mathsf{inf}}(G_0)$\, defined as the supremum of all cross n
umbers of minimal (by inclusion) zero-sum sequences\, the supremum of all
cross numbers of zero-sum free sequences (sequences having no zero-sum sub
sequence)\, and the infimum of all cross numbers of nontrivial zero-sum se
quences. The first two of these constants have appeared in the literature
before\, but the third is newly introduced here. \n\nIn the first part of
this two part talk\, it was shown that factorization of zero-sum sequences
can be very ill-behaved when $\\mathsf K_{\\mathsf{inf}}(G_0)=0$. In this
second part\, we consider what happens when $\\mathsf K_{\\mathsf{inf}}(G
_0)>0$\, specifically in the setting of infinite abelian torsion groups wi
th finite total rank. In this setting\, the first two cross number constan
ts $\\mathsf K(G_0)$ and $\\mathsf k(G_0)$ are always finite. Assuming $\\
delta:=\\mathsf K_{\\mathsf{inf}}(G_0)>0$\, we then obtain a general upper
bound for the catenary degree $$\\mathsf c(G_0)\\leq \\max\\{2\\delta^{-1
}\\mathsf k(G_0)+1\, \\quad 2\\delta^{-1}\\mathsf K(G_0)\\}.$$ In particul
ar\, this implies that both the set of successive distances $\\Delta(G_0)$
and catenary degree are always finite under these circumstances\, with ex
plicit concrete upper bounds. Moreover\, our upper bound on the catenary d
egree is tight\, meaning there are infinite families of subsets $G_0\\subs
eteq G$ for which equality holds above. In addition\, for the special case
of quasi-cyclic groups\, we are able to partially characterize what subse
ts $G_0$ with $\\mathsf K_{\\mathsf{inf}}(G_0)>0$ look like and use this t
o give a lower bound for the elasticities $\\rho_k(G_0)$. Combined with th
e upper bound on the catenary degree\, this yields a structural descriptio
n of the possible refactorization lengths of a product of $k$ irreducibles
. This is joint work with Alfred Geroldinger and Guoqing Wang.\n
LOCATION:https://researchseminars.org/talk/CANT2026/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Gabdullin (University of Illinois at Urbana-Champaign)
DTSTART:20260713T150000Z
DTEND:20260713T152500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/13
DESCRIPTION:Title: Moments of the shifted prime divisor function\nby Mikhail Gabdullin
(University of Illinois at Urbana-Champaign) as part of Combinatorial and
additive number theory seminar (CANT 2026)\n\nLecture held in Science Cent
er in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $\\omega^*(n)
= \\{d|n: d=p-1\, \\mbox{$p$ is a prime}\\}$ denote the ``shifted prime d
ivisor'' function. It is easy to see that $\\sum_{n\\leq x}\\omega^*(n)=x\
\log\\log x+O(x)$\, similar to the average value of $\\omega(n)$\, the num
ber of prime divisors of $n$. We confirm a recent conjecture of Fan and Po
merance by proving that\, for each integer $k\\geq2$\, $\n\\qquad \\sum_{n
\\leq x}\\omega^*(n)^k \\asymp x(\\log x)^{2^k-k-1}\,\n$ \nwhere the impli
ed constant may depend only on $k$. The proof relies on a combinatorial id
entity for the least common multiple\, viewed as a multiplicative analogue
of the inclusion-exclusion principle\, together with the theory of multip
licative functions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alfred Geroldinger (University of Graz\, Austria)
DTSTART:20260713T153000Z
DTEND:20260713T155500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/14
DESCRIPTION:Title: On factorizations of zero-sum sequences over abelian torsion groups I\nby Alfred Geroldinger (University of Graz\, Austria) as part of Combina
torial and additive number theory seminar (CANT 2026)\n\nLecture held in S
cience Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $G
$ be an additive abelian group and let $G_0\\subseteq G$ be a subset. A ze
ro-sum sequence over $G_0$ is an unordered string of terms from $G_0$ (rep
etition of terms allowed) such that the sum of terms is $0$. The study of
zero-sum sequences dates back over 60 years\, and while they have often be
en studied for purely combinatorial interest\, the original motivation was
due to connections with factorization in rings of integers in algebraic n
umber fields. In the last few decades\, the connection between factorizati
ons of zero-sum sequences and factorization of elements in rings of intege
rs was made more precise and extended into much more general algebraic set
tings. This then allows the algebraic structure of factorization to be stu
died via combinatorial properties of zero-sum sequences. We briefly review
this connection\, making all notions concrete\, and then turn our focus t
o the combinatorial part. In the past\, there has been much focus on finit
e groups\, and more recently\, on subsets of finitely generated groups. Ho
wever\, very little was known in the case of non-finitely generated abelia
n groups. In part\, this is because common invariants used to study factor
ization\, such as the Davenport Constant\, are no longer guaranteed to be
finite. In order to better understand factorization in the setting of infi
nite abelian torsion groups\, we introduce a new technique measuring the s
ize of a sequence not by the number of its terms but rather by its cross n
umber\, $\\sum_{i=1}^{\\ell} \\frac{1}{\\text{\\rm ord} (g_i)}$\, where th
e $g_i\\in G_0 \\subseteq G$ are the terms in the sequence. Cross numbers
have previously been used almost solely for finite groups. In order to ada
pt their use into the infinite torsion group setting\, we need to introduc
e a new invariant\, $\\mathsf K_{\\mathsf{inf}}(G_0)$\, defined as the inf
imum of all cross numbers of nontrivial zero-sum sequences with terms from
$G_0$. This then sets up dichotomy between when $\\mathsf K_{\\mathsf{inf
}}(G_0)=0$ and when $\\mathsf K_{\\mathsf{inf}}(G_0)>0$. In this first par
t of two talks\, we focus on when $\\mathsf K_{\\mathsf{inf}}(G_0)=0$\, an
d show that factorization of zero-sum sequences can be very ill-behaved un
der this assumption. In the follow-up talk\, we then instead consider when
$\\mathsf K_{\\mathsf{inf}}(G_0)>0$ and see that this instead guarantees
that factorization must be well-behaved\, as measured by the finiteness of
several commonly factorization metrics. This is joint work with David J.
Grynkiewicz and Guoqing Wang.\n
LOCATION:https://researchseminars.org/talk/CANT2026/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sinan Gunturk (New York University)
DTSTART:20260713T173000Z
DTEND:20260713T175500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/16
DESCRIPTION:Title: Exponential sums and a conjecture involving quantization of bandlimited
functions\nby Sinan Gunturk (New York University) as part of Combinato
rial and additive number theory seminar (CANT 2026)\n\nLecture held in Sci
ence Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nSigma-de
lta modulation is a classical method for oversampled coarse quantization w
hich enables approximation of bandlimited functions (e.g. audio signals) a
t high sampling rates despite using only two fixed levels to round each sa
mple. In the basic form of this method (the "first order" case)\, the appr
oximation rate is $\\lambda^{-1}$ in the uniform norm where $\\lambda$ den
otes the oversampling ratio\, but the pointwise error has been shown to de
cay at least at the rate $\\lambda^{-4/3+\\epsilon}$ under generic conditi
ons. Meanwhile\, a long-standing folklore conjecture based on numerical si
mulations predicts square-root cancellation "on average"\, i.e. approximat
ion rate of order $\\lambda^{-3/2+\\epsilon}$. We disprove the conjecture
for the Besicovitch norm\, utilizing certain exponential sums of bandlimit
ed phase. Joint work with Maksym Radziwill.\n
LOCATION:https://researchseminars.org/talk/CANT2026/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Shkredov (Purdue University)
DTSTART:20260713T180000Z
DTEND:20260713T185000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/17
DESCRIPTION:Title: On Korobov's optimal coefficients\nby Ilya Shkredov (Purdue Universi
ty) as part of Combinatorial and additive number theory seminar (CANT 2026
)\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor
).\n\nAbstract\nLet $p$ be a prime number\, $d$ be a positive integer\, an
d $M\\ge 1$ be a real parameter. A tuple $(a_1\,\\dots\, a_d) \\in \\mathb
f{F}^d_p$ is called a tuple of (Korobov) {\\it optimal coefficients} if\,
for any nonzero $x\\in \\mathbf{F}_p$\, the inequality$$\n x|a_1 x| \\dots
|a_d x| \\ge \\frac{p^d}{M} \n$$ holds. \n These famous coefficients ari
se naturally in numerical integration problems. Namely\, if a tuple $(a_1
\, \\dots\, a_d)$ satisfying the inequality is found\, then any function $
f:[0\,1]^d \\to \\mathbf{R}$ can be integrated using the formula $$\n\\lef
t| \\int_{[0\,1]^d} f(x)\\\,dx - \\frac{1}{p} \\sum_{x=1}^{p} f\\left(\\fr
ac{a_1 x}{p}\, \\dots\, \\frac{a_d x}{p} \\right) \\right| \\ll \\frac{M\\
cdot \\mathrm{V}(f)}{p} \\\,\,\n$$\n where $\\mathrm{V}(f)$ is the Hardy--
Krause variation of the function $f$. \nKorobov proved that the case $M=O(
(\\log p)^{d-1})$ is always realizable\, whereas the special case $d=1$\,
$M=O(1)$ is equivalent to the well-known Zaremba conjecture.\nFor $d>1$ an
d arbitrary $M$\, only a few results are known. In our talk\, we will prov
ide an overview of the problems in this area and describe recent advances
and connections to other topics in number theory.\n
LOCATION:https://researchseminars.org/talk/CANT2026/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (Stanford University)
DTSTART:20260713T190000Z
DTEND:20260713T195000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/18
DESCRIPTION:Title: Primitive sets and von Mangoldt chains: Erdős #1196 and beyond\nby
Jared Duker Lichtman (Stanford University) as part of Combinatorial and ad
ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center
in the CUNY Graduate Center (4th floor).\n\nAbstract\nA set of integers i
s primitive if no number in the set divides another. We introduce a new me
thod for bounding Erdős sums of primitive sets\, suggested from output of
GPT-5.4 Pro\, based on Markov chains with von Mangoldt weights. The metho
d leads to a host of applications\, yet seems to have been overlooked by t
he prior literature since Erdős' seminal 1935 paper. As applications\, we
prove two 1966 conjectures of Erdős-Sárközy-Szemerédi\, on primitive
sets of large numbers (#1196) and on divisibility chains (#1217). The meth
od also provides a short proof of the Erdős Primitive Set Conjecture (#16
4)\, as well as the related claim that 2 is an ``Erdős-strong'' prime. Mo
reover\, the method resolves a revised form of the Banks-Martin conjecture
\, which has long been viewed as a unifying ``master theorem'' for the are
a. Joint work with B. Alexeev\, K. Barreto\, Y. Li\, L. Price\, J. I. Shah
\, Q. Tang\, and T. Tao.\n
LOCATION:https://researchseminars.org/talk/CANT2026/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Miller’s REU: Probability And Number THeory (Williams Col
lege)
DTSTART:20260713T200000Z
DTEND:20260713T205000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/19
DESCRIPTION:Title: Recent advances in generalized MSTD problems and Zeckendorf games\nb
y Steven Miller’s REU: Probability And Number THeory (Williams College)
as part of Combinatorial and additive number theory seminar (CANT 2026)\n\
nLecture held in Science Center in the CUNY Graduate Center (4th floor).\n
\nAbstract\nWe report on two areas studied this summer in Miller's REU: Ge
neralized\nMSTD Problems and Zeckendorf Games. \n\n1. A finite integer sub
set $A \\subseteq\n\\mathbb{Z}$ is classified as a More Sums Than Differen
ces (MSTD\, or\nsum-dominant) set when it produces strictly more pairwise
sums than\ndifferences\, satisfying $|A+A| > |A-A|$. Motivated by the stru
ctural\ndensity of these integer sets\, we generalize this phenomenon to s
ubsets\n$A$ of a finite group $G$ by comparing the cardinality of the prod
uct set\n$AA$ against the quotient set $AA^{-1}$. To evaluate global group
\nbehavior\, we analyze the weighted difference across all possible subset
s\,\ndefined as $$W(G) = \\sum_{A \\subseteq G} (|AA| - |AA^{-1}|).$$ Usin
g a\ncombination of combinatorial techniques\, graph theory\, and represen
tation\ntheory\, we prove that $W(G)$ is strictly negative for all finite
abelian\ngroups—establishing them as inherently quotient-dominant—and
we successfully extend these structural findings to characterize select\nn
on-abelian groups. \n\n2. Zeckendorf proved every integer can be written u
niquely as a sum of\nnon-adjacent Fibonacci numbers $\\{F_n\\}$. Using the
Fibonacci recurrence\,\nMiller created the Zeckendorf game. Starting with
$n$ copies of $F_1$\, a\nplayer either replaces a copy of $F_i$ and $F_{i
-1}$ with $F_{i+1}$\, or\nsplits two copies of $F_i$ into $F_{i+1}$ and $F
_{i-1}$ (with $F_2$\nsplitting to $F_3$ and $F_1$). All games terminate in
the Zeckendorf\ndecomposition of $n$\; whomever moves last wins. A non-co
nstructive proof\nexists that Player Two has a winning strategy for all $n
> 2$. We discuss\ncurrent work on a variety of generalizations\, includin
g binary\ndecompositions\, first to reach the largest summand wins\, and h
igher\ndimensional analogues.\n
LOCATION:https://researchseminars.org/talk/CANT2026/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Miller (Williams College)
DTSTART:20260713T210000Z
DTEND:20260713T213000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/20
DESCRIPTION:Title: Problem Session\nby Steve Miller (Williams College) as part of Combi
natorial and additive number theory seminar (CANT 2026)\n\nLecture held in
Science Center in the CUNY Graduate Center (4th floor).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johann Thiel (New York College of Technology (CUNY)
DTSTART:20260714T130000Z
DTEND:20260714T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/21
DESCRIPTION:Title: Generating functions for maximally expanded α-trees\nby Johann Thie
l (New York College of Technology (CUNY) as part of Combinatorial and addi
tive number theory seminar (CANT 2026)\n\nLecture held in Science Center i
n the CUNY Graduate Center (4th floor).\n\nAbstract\nWe construct generati
ng functions whose coefficients enumerate certain directed planar trees kn
own as maximally expanded $\\alpha$-trees. We show that the number of such
trees can be expressed as an integer linear combination of Catalan number
s. This is joint work with David M. Bradley.\n
LOCATION:https://researchseminars.org/talk/CANT2026/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Konyagin (Russia)
DTSTART:20260714T133000Z
DTEND:20260714T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/22
DESCRIPTION:Title: On Sidon sets with squares\, cubes and quartics in short intervals\n
by Sergei Konyagin (Russia) as part of Combinatorial and additive number t
heory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Gr
aduate Center (4th floor).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Renling Jin (College of Charleston)
DTSTART:20260714T143000Z
DTEND:20260714T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/23
DESCRIPTION:Title: Three-in-one in Ramsey theory\nby Renling Jin (College of Charleston
) as part of Combinatorial and additive number theory seminar (CANT 2026)\
n\nLecture held in Science Center in the CUNY Graduate Center (4th floor).
\n\nAbstract\nThere are three fundamental theorems in Ramsey theory: Ramse
y's theorem\, van der Waerden's theorem\, and Hindman's theorem. Milliken-
Taylor proved a result that simultaneously generalizes Ramsey's theorem an
d Hindman's theorem. Later\, Bergelson--Hindman and Samet--Tsaban \n estab
lished two distinct theorems\, each providing a simultaneous generalizati
on \n of Ramsey's theorem and van der Waerden's theorem in two different
ways. Using a newly \n developed method of iterated extensions\, we prove-
-pending verification--a theorem that \n simultaneously generalizes all th
ree classical results--Ramsey's theorem\, Hindman's theorem\, and van der
Waerden's theorem. Moreover\, our theorem subsumes both the Bergelson--Hin
dman and the Samet--Tsaban generalizations. Joint work with Mauro Di Nasso
.\n
LOCATION:https://researchseminars.org/talk/CANT2026/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Scott Chapman (Sam Houston State University)
DTSTART:20260714T150000Z
DTEND:20260714T152500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/24
DESCRIPTION:Title: A surprising characterization of unique factorization domains\nby Sc
ott Chapman (Sam Houston State University) as part of Combinatorial and ad
ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center
in the CUNY Graduate Center (4th floor).\n\nAbstract\nA surprising charac
terization of unique factorization domains \\\\\nAbstract: & We address so
me recent work on the generalization of the UFD propery which has pointed
back to an open problem first mentioned in a paper by myself\, Dan Anderso
n\, Muhammad Zafrullah\, and Franz Halter-Koch (Criteria for unique factor
ization in integral domains\, J. Pure Appl. Algebra 127(1998)\, 205--218)\
, which we abbreviate as ACHKZ. Fix a positive integer $n>1$. Call an atom
ic integral domain $D$ quasi-$n$-factorial if\, for any irreducible elemen
ts \n$x_1\, \\ldots \, x_n\, y_1\, \\ldots \, y_n$\, the equality\n$x_1\\c
dots x_n=y_1\\cdots y_n$ implies that $x_i=u_iy_{\\sigma(i)}$ for some uni
t $u_i$ and permutation $\\sigma$ of $\\{1\,\\ldots \,n\\}$. Further\, $D$
is length-factorial if it is quasi-$n$-factorial for all $n>1$. Jim Coyke
ndall and William W. Smith showed in 2011 the surprising result that an at
omic monoid is a UFD if and only if it is length-factorial. This allows on
e to alter the classic definition of a UFD. in a surprising manner. The au
thors in ACHKZ offer examples of monoids which are quasi-$n$-factorial for
specific $n$\, but are not factorial. They offer no such example of an in
tegral domain. Hence\, the Coykendall-Smith result makes the following pro
blem explored in ACHKZ all the more relevant. Open Problem: Does there exi
st an atomic integral domain $D$ which is quasi-$n$-factorial for some $n>
1$\, but not factorial?\n
LOCATION:https://researchseminars.org/talk/CANT2026/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akos Magyar (University of Georgia)
DTSTART:20260714T153000Z
DTEND:20260714T165500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/25
DESCRIPTION:Title: Almost primes solutions to forms of odd degrees in many variables\nb
y Akos Magyar (University of Georgia) as part of Combinatorial and additiv
e number theory seminar (CANT 2026)\n\nLecture held in Science Center in t
he CUNY Graduate Center (4th floor).\n\nAbstract\nLet $\\mathcal{F}=\\{f_1
\,\\ldots\,f_R\\}$ be a family of forms of odd degrees at most $d$ in $s$
variables. We study the solutions to the diophantine system: $f_1(\\mathbf
{x})=\\ldots=f_R(\\mathbf{x})=0$ of the form $x_i=y_ip_i$ with $|y_i|\\leq
Y_\\mathcal{F}$ and $p_i$ being a prime for all $i\\in [s]$ inside the bo
x $[-N\,N]^s$\, for large $N$. We show that if the number of variables $s$
is sufficiently large with respect to the parameters $R$ and $d$\, then t
here are at least $C_\\mathcal{F} N^{s-D}/(\\log\\\,N)^s$ such solutions f
or some constants $C_\\mathcal{F}>0$ and $D\\in\\mathbb{N}$\, with $D$ dep
ending only on the initial parameters $R$ and $d$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20260714T173000Z
DTEND:20260714T182000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/26
DESCRIPTION:Title: Two topics in combinatorial number theory\nby Carl Pomerance (Dartmo
uth College) as part of Combinatorial and additive number theory seminar (
CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center (
4th floor).\n\nAbstract\nThe first topic: In a paper with Erd\\H os from 4
0 years ago\,\nwe considered the set of residues $a \\bmod n$ where\n$a^{n
-1} \\equiv 1 \\pmod n$.\nIf $n$ is composite\, these are the bases for wh
ich $n$ is a pseudoprime.\nRecently\, Lenstra asked me about the set of re
sidues $a \\bmod n$\nwhere $a^n \\equiv 1 \\pmod n$\, which is related to
a problem he is\nworking on about conditions that ensure a ring is commuta
tive.\nSome of the methods from the old paper were useful in the new\nprob
lem\, but not all. I will discuss the more general problem of subgroups of
the multiplicative group mod $n$. The second topic: I will discuss\nsome
old and new problems on coprime matchings: These are perfect\nmatchings be
tween two equally numerous sets of integers\, where each matched pair is r
elatively prime. Some examples: Given two intervals\nof $n$ consecutive in
tegers is there a coprime matching between them?\nIf both intervals are $\
\{1\,2\,\\dots\,n\\}$\, how many such matchings are\nthere? For a positive
integer $n$\, is there a coprime matching between\nthe set $D(n)$ of divi
sors of $n$ and an interval of $D(n)$ consecutive\nintegers? This last pro
blem reflects joint work with Nathan McNew.\n
LOCATION:https://researchseminars.org/talk/CANT2026/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksym Radziwill (New York University)
DTSTART:20260714T183000Z
DTEND:20260714T185500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/27
DESCRIPTION:Title: Exponential sums over primes\nby Maksym Radziwill (New York Universi
ty) as part of Combinatorial and additive number theory seminar (CANT 2026
)\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor
).\n\nAbstract\nA classical result of Vinogradov shows that\, for any $\\a
lpha$ with $$\n\\Big | \\alpha - \\frac{a}{q} \\Big | \\leq \\frac{1}{q^2}
\\ \, \\ q \\leq x^{1/2}\,\n$$ \nand for any $\\varepsilon > 0$\, we have
\, $$\n\\Big | \\sum_{p \\leq x} e^{2\\pi i \\alpha p} \\Big | \\leq C(\\v
arepsilon) x^{\\varepsilon} \\cdot \\Big ( \\frac{x}{\\sqrt{q}} + x^{4/5}
\\Big ).\n$$ \nwith $C(\\varepsilon) > 0$ a constant depending only on $\\
varepsilon$.\nThis has resisted improvements for the past 80 years\, beyon
d\nrefinements to the $x^{\\varepsilon}$ term. The $x / \\sqrt{q}$ term ca
nnot be improved without eliminating the existence of a Siegel zero. I'll
discuss joint work with James Maynard and Mayank Pandey\, in which we redu
ce the exponent $4/5$ appearing in $x^{4/5}$ to $19/24$\, which should hav
e various applications to additive problems related to primes.\n
LOCATION:https://researchseminars.org/talk/CANT2026/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishnaswami Alladi (University of Florida)
DTSTART:20260714T190000Z
DTEND:20260714T195000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/28
DESCRIPTION:Title: Duality between prime factors and prime numbers in arithmetic progressio
ns\nby Krishnaswami Alladi (University of Florida) as part of Combinat
orial and additive number theory seminar (CANT 2026)\n\nLecture held in Sc
ience Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nIn 1977
\, I noticed a duality between the largest and smallest\n prime factors of
the integers involving the Mobius function\, and used this to establish
the following result as a consequence of the Prime Number Theorem\n for A
rithmetic Progressions: \n If $k$ and $\\ell$ are positive\n integers\, wi
th $1\\le \\ell\\le k$ and $(\\ell\, k)=1$\, then $$ \n \\sum_{n\\ge 2\,
\\\, p(n)\\equiv\\ell(mod\\\,k)}\\frac{\\mu(n)}{n}=\\frac{-1}{\\phi(k)}\,
$$ where $\\mu(n)$ is the Mobius function\, $p(n)$ is the\n smallest prime
factor of $n$\, and $\\phi(k)$ is the Euler function. In the last decade
\, several authors have obtained analogues of (1) in the setting of algebr
aic number fields by using the Chebotarev Density Theorem. Also in 1977\,
I proved higher order duality identities involving the $k$-th largest and
smallest prime factors\, facilitated by the Mobius function and $\\omega(
n)$\, the number of distinct prime factors of $n$. In this talk we will ex
ploit the second order duality between the second largest prime factor and
the smallest prime factor\, to show that if $\\ell$ and $k$ are as above\
, then $$ \n \\sum_{n\\ge 2\,\\\, p(n)\\equiv\\ell(mod\\\,k)}\\frac{\\mu(n
)\\omega(n)}{n}=0. \n $$ The proof of (2) is more complicated owing to the
weight $\\omega(n)$\, and also because it relies on the distribution of
the second largest prime factor which is more subtle compared to the dist
ribution of the largest prime factor. All results are established quantita
tively. This is joint work with my PhD student Jason Johnson. Recently\,
another PhD student of mine\, Sroyon Sengupta\, has extended the Alladi-J
ohnson results to algebraic number fields using the Chebotarev Density The
orem. \nTowards the end of the talk\, we will briefly mention further join
t work with Sengupta on consequences of such dualities involving the $k-th
$ largest and smallest prime factors\, when $k\\ge 3$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Andrews (Pennsylvania State University)
DTSTART:20260714T200000Z
DTEND:20260714T205000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/29
DESCRIPTION:Title: The mystery of two-color partitions with distinct parts\nby George A
ndrews (Pennsylvania State University) as part of Combinatorial and additi
ve number theory seminar (CANT 2026)\n\nLecture held in Science Center in
the CUNY Graduate Center (4th floor).\n\nAbstract\nWe shall present some o
ld and some new results about two-color partitions with distinct parts.
In the midst of our exploration\, a power series arises that seems to be "
semi-lacunary." What is going on anyway? The answers to this and other
mysteries will be provided. Joint work with M. El Bachraoui.\n
LOCATION:https://researchseminars.org/talk/CANT2026/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishnaswami Alladi (University of Florida)
DTSTART:20260714T210000Z
DTEND:20260714T213000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/30
DESCRIPTION:Title: Problem session\nby Krishnaswami Alladi (University of Florida) as p
art of Combinatorial and additive number theory seminar (CANT 2026)\n\nLec
ture held in Science Center in the CUNY Graduate Center (4th floor).\nAbst
ract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sukumar Das Adhikar (Ramakrishna Mission Vivekananda Educational a
nd Research Institute\, India)
DTSTART:20260715T130000Z
DTEND:20260715T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/31
DESCRIPTION:Title: A pearl of number theory: Some old and new applications\nby Sukumar
Das Adhikar (Ramakrishna Mission Vivekananda Educational and Research Inst
itute\, India) as part of Combinatorial and additive number theory seminar
(CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center
(4th floor).\n\nAbstract\nAfter stating the classical van der Waerden's t
heorem\, and a brief discussion of its relation with some early Ramsey-typ
e theorems\,\nwe go through some old and new applications of the theorem.
We shall also see some open questions in Ramsey Theory.\n
LOCATION:https://researchseminars.org/talk/CANT2026/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jinhui Fang (Nanjing Normal University\, Nanjing\, China)
DTSTART:20260715T133000Z
DTEND:20260715T135500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/32
DESCRIPTION:Title: Minimal asymptotic bases related to G-adic sequences\nby Jinhui Fang
(Nanjing Normal University\, Nanjing\, China) as part of Combinatorial an
d additive number theory seminar (CANT 2026)\n\nLecture held in Science Ce
nter in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $A$ be a se
t of nonnegative integers and $h\\ge 2$. The set $A$ is defined as an asym
ptotic basis of order $h$ if all sufficiently large integers $n$ can be ex
pressed as the sum of $h$ elements taken from $A$. Such $A$ is further def
ined as \\emph{minimal} if no proper subset of $A$ is an asymptotic basis
of order $h$. In 1974\, Nathanson explicitly constructed a minimal asympto
tic basis of order $2$ by using binary representations. In 2022\, Nathanso
n constructed a new class of minimal asymptotic bases of order $h$ based o
n the $\\mathcal{G}$-adic sequence\, where a $\\mathcal{G}$-adic sequence
$\\mathcal{G}=\\{g_i\\}_{i=0}^{\\infty}$ is a strictly increasing sequence
of positive integers such that $g_0=1$ and $g_{i-1}$ divides $g_i$ for al
l $i\\ge 1$. Recently\, we improve the above result.\n
LOCATION:https://researchseminars.org/talk/CANT2026/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ivan V. Morozov (City College (CUNY))
DTSTART:20260715T140000Z
DTEND:20260715T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/33
DESCRIPTION:Title: On quotients of a more general theorem of Wilson\nby Ivan V. Morozov
(City College (CUNY)) as part of Combinatorial and additive number theory
seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduat
e Center (4th floor).\n\nAbstract\nThe basis of this work is a corollary a
nd generalization of Wilson’s theorem\, $(-1)^{k}k!(n-k-1)!\\equiv -1\\p
mod{n}$ iff $n$ is non-composite\, for $0\\leq k\\leq n-1$. This corollary
generates many more quotients than those already generated by Wilson’s
theorem\, and we derive how they relate to each other and build on the est
ablished properties of the original quotients. The main results are expres
sions for sums of these quotients\, modular congruences that extend the re
sults of Lehmer\, and generating functions. In addition\, a solution will
be provided for an open problem raised in CANT 2025 by Brian Hopkins regar
ding a combinatorial proof for the partition identity $p(a\,3)+p(b\,3)=p(c
\,3)$\, where $a$\, $b$\, and $c$ comprise a Pythgagorean triple.\n
LOCATION:https://researchseminars.org/talk/CANT2026/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey C. Lagarias (University of MIchigan)
DTSTART:20260715T143000Z
DTEND:20260715T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/34
DESCRIPTION:Title: The Collatz problem: Progress and perspectives\nby Jeffrey C. Lagari
as (University of MIchigan) as part of Combinatorial and additive number t
heory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Gr
aduate Center (4th floor).\n\nAbstract\nThe Collatz problem concerns the i
teration of the map $C(n) = n/2$ if $n$ is even\; $C(n) = 3n + 1$ if $n$ i
s odd\, on the positive integers. It asks whether the integer 1 is reached
for all starting\nvalues $n$. This talk surveys some history and recent p
rogress towards the Collatz Problem. It\noffers some perspectives on its d
ifficulty.\n
LOCATION:https://researchseminars.org/talk/CANT2026/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College and CUNY Graduate Center)
DTSTART:20260715T153000Z
DTEND:20260715T162000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/35
DESCRIPTION:Title: Three problems in additive number theory\nby Mel Nathanson (Lehman C
ollege and CUNY Graduate Center) as part of Combinatorial and additive num
ber theory seminar (CANT 2026)\n\nLecture held in Science Center in the CU
NY Graduate Center (4th floor).\n\nAbstract\nThis will be an introduction
to three (possibly new) problems in additive number theory. The first conc
erns the range and frequencies of the sizes of sumsets of finite sets of i
ntegers. The second considers the sets $H$ of integers such that there exi
sts an increasing sequence $(A_i)_{i=1}^{\\infty} A_i$ of sets of integers
such that $h \\in H$ if and only if $h\\bigcap_{i=1}^{\\infty} A_i = \\bi
gcap_{i=1}^{\\infty} hA_i$. The third asks about the possible sizes of $h$
-bases for $n$ for finite sets of integers that contain at least one negat
ive integer.\n
LOCATION:https://researchseminars.org/talk/CANT2026/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20260715T173000Z
DTEND:20260715T175500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/36
DESCRIPTION:Title: Nathanson’s triangular gap question\nby Steven Senger (Missouri St
ate University) as part of Combinatorial and additive number theory semina
r (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Cente
r (4th floor).\n\nAbstract\nMel Nathanson recorded the size distribution o
f iterated sumsets of four natural numbers chosen from a large interval of
integers. He observed that the most frequent sizes were not evenly distri
buted\, but had gaps between them\, and that these gaps were consecutive t
riangular numbers. We explain this phenomenon in full detail.\n
LOCATION:https://researchseminars.org/talk/CANT2026/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alisa Sedunova (Purdue University)
DTSTART:20260715T180000Z
DTEND:20260715T182500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/37
DESCRIPTION:Title: Euler-Kronecker constants of maximal real cyclotomic subfields and Kumme
r’s conjecture\nby Alisa Sedunova (Purdue University) as part of Com
binatorial and additive number theory seminar (CANT 2026)\n\nLecture held
in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nTh
e Euler–Kronecker constant of a number field $K$ is the ratio of the con
stant and the residue of the Laurent series of the Dedekind zeta function
at $s = 1$. We study the distribution of the Euler–Kronecker constant $\
\gamma_q^+$ of the maximal real subfield $\\mathbb{Q}(\\zeta_q)^+$ as $q$
ranges over the primes. Further\, we consider the distribution of $\\gamma
_q^+ - \\gamma_q$\, with $\\gamma_q$ the Euler–Kronecker constant of $\\
mathbb{Q}(\\zeta_q)$ and show how it is connected with Kummer’s conjectu
re\, which predicts the asymptotic growth of the relative class number of
$\\mathbb{Q}(\\zeta_q)$. We improve\, for example\, the known results on t
he bounds on average for the Kummer ratio and we prove analogous sharp bou
nds for $\\gamma_q^+ - \\gamma_q$. The methods employed are partly inspire
d by those used by Granville (1990) and Croot and Granville (2002) to inve
stigate Kummer’s conjecture\, that predicts the asymptotic growth of the
relative class number of prime cyclotomic fields. We substantially improv
e the known bounds of Kummer’s ratio under three scenarios: no Siegel ze
ro\, presence of Siegel zero and assuming the Riemann Hypothesis for the D
irichlet $L$-series attached to odd characters only. \nThe talk is based o
n joint papers with A. Languasco\, P. Moree\, N. Kandhil and S. Saad Eddin
.\n
LOCATION:https://researchseminars.org/talk/CANT2026/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Borisov (Binghamton University)
DTSTART:20260715T183000Z
DTEND:20260715T185500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/38
DESCRIPTION:Title: A structure sheaf for Kirch topology on N\nby Alexander Borisov (Bin
ghamton University) as part of Combinatorial and additive number theory se
minar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate C
enter (4th floor).\n\nAbstract\nKirch topology on $\\mathbb N$ goes back t
o a 1969 paper of Kirch. It can be defined by a basis of open sets that co
nsists of all infinite arithmetic progressions $a+d\\mathbb N_0$\, such th
at $\\gcd(a\,d)=1$ and $d$ is square-free. It is Hausdorff\, connected\, a
nd locally connected. One can hope that in the classical imperfect analogy
between arithmetic and geometry this can serve as an arithmetic analog of
the usual topology on $\\mathbb C$. However\, the usual topology on $\\ma
thbb C$ comes with a structure sheaf of complex-analytic functions. As far
as I know\, no analog for Kirch topology has been proposed before me. I b
elieve that I have stumbled upon just such a thing\, more by accident than
by a conscious effort: locally LIP functions. These are functions from Ki
rch-open sets to $\\mathbb Z$ such that for every point in the domain ther
e is a Kirch-open neighborhood on which the function is "locally integer p
olynomial" (LIP): its interpolation polynomial on every finite set has int
eger coefficients. I will explain why this seems to be a natural object\,
what I know about it\, and what I hope to achieve.\n
LOCATION:https://researchseminars.org/talk/CANT2026/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wladimir Pribitkin (College of Staten Island and CUNY Graduate Cen
ter)
DTSTART:20260715T190000Z
DTEND:20260715T192500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/39
DESCRIPTION:Title: Simple upper bound for the power partition function\nby Wladimir Pri
bitkin (College of Staten Island and CUNY Graduate Center) as part of Comb
inatorial and additive number theory seminar (CANT 2026)\n\nLecture held i
n Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nRei
magining Siegel's method\, we shall produce a rather easy proof of a surpr
isingly good upper bound on the number of partitions of a positive integer
into perfect $r$th powers\, where $r \\ge 1$. If time permits\, we shall
present a generalization pertaining to partitions into perfect powers of t
erms in an arithmetic progression.\n
LOCATION:https://researchseminars.org/talk/CANT2026/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Dion Wooley (Purdue University)
DTSTART:20260715T193000Z
DTEND:20260715T202000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/40
DESCRIPTION:Title: Strong paucity in systems of diagonal equations\nby Trevor Dion Wool
ey (Purdue University) as part of Combinatorial and additive number theory
seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduat
e Center (4th floor).\n\nAbstract\nLet $k$ be a natural number with $k\\ge
2$\, and let $\\varepsilon>0$. We consider the number\n$V_k^*(P)$ of inte
gral solutions of the system of simultaneous Diophantine equations $$x_1^{
2j-1}+\\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\\ldots +y_{k+1}^{2j-1}\\quad (1\
\le j\\le k).$$ with $1\\le x_i\,y_i\\le P$ $(1\\le i\\le k+1)$. Writing $
L_k^*(P)$ for the number of diagonal solutions with \n$\\{x_1\,\\ldots \,x
_{k+1}\\}=\\{y_1\,\\ldots \,y_{k+1}\\}$\, so that $L_k^*(P)\\sim (k+1)!P^{
k+1}$\, we prove that $$V_k^*(P)-L_k^*(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsil
on}.$$ This establishes a strong paucity result improving on earlier work
of Brüdern and Robert. Time permitting\, we describe analogous results fo
r related problems.\n
LOCATION:https://researchseminars.org/talk/CANT2026/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Filaseta (University of South Carolina)
DTSTART:20260715T203000Z
DTEND:20260715T205500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/41
DESCRIPTION:Title: On the factorization of a sum of cyclotomic polynomials\nby Michael
Filaseta (University of South Carolina) as part of Combinatorial and addit
ive number theory seminar (CANT 2026)\n\nLecture held in Science Center in
the CUNY Graduate Center (4th floor).\n\nAbstract\nIn 2000\, Charles Nico
l conjectured that for $n$ and $m$ integers with $n > m >1$\, the sum $\\P
hi_{n}(x)+\\Phi_{m}(x)$ is a product of distinct cyclotomic polynomials an
d either a constant or an irreducible non-cyclotomic polynomial. Little pr
ogress has been made on this conjecture since then. In this talk\, we disc
uss recent joint work with Lilit Martirosyan and London Swan\, where\, in
particular\, we show that for primes $p$\, $q$ and $\\ell$ with $p > q > \
\ell$ and a non-negative integer $r$\, the sum $\\Phi_{\\ell^{r} p}(x)+\\P
hi_{\\ell^{r} q}(x)$ has this property and determine precisely the cycloto
mic polynomials dividing the sum.\n
LOCATION:https://researchseminars.org/talk/CANT2026/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Senger (Missouri State University)
DTSTART:20260715T210000Z
DTEND:20260715T213000Z
DTSTAMP:20260710T091017Z
UID:CANT2026/42
DESCRIPTION:Title: Problem Session\nby Steve Senger (Missouri State University) as part
of Combinatorial and additive number theory seminar (CANT 2026)\n\nLectur
e held in Science Center in the CUNY Graduate Center (4th floor).\nAbstrac
t: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arindam Biswas (Polynom Research\, Paris\, France)
DTSTART:20260716T130000Z
DTEND:20260716T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/43
DESCRIPTION:Title: Asymptotic approximate groups in virtually nilpotent groups\nby Arin
dam Biswas (Polynom Research\, Paris\, France) as part of Combinatorial an
d additive number theory seminar (CANT 2026)\n\nLecture held in Science Ce
nter in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet \\(G\\) be
a group and let \\(A\\subseteq G\\) be a non-empty subset. For\n\\(r\,l\\i
n\\mathbb N\\)\, \\(A\\) is said to be an asymptotic\n\\((r\,l)\\)-approxi
mate group if there exists \\(h_0\\in\\mathbb N\\) such that\,\nfor every
\\(h\\ge h_0\\)\, there is a set \\(X_h\\subseteq G\\) with\n\\(|X_h|\\le
l\\) and\n$A^{rh}\\subseteq X_hA^h.$\nWe study this property for subsets o
f virtually nilpotent groups and show that\nevery finite non-empty symmetr
ic subset of a virtually nilpotent group is an\nasymptotic approximate gro
up. More generally\, the same conclusion holds for finite\nsets whose powe
rs contain a symmetric word ball of radius comparable to \\(h\\). In the s
etting of infinite sets\, we show a restricted nonabelian analogue of the
abelian semilinear-set theorem.\n
LOCATION:https://researchseminars.org/talk/CANT2026/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Oxford University\, UK)
DTSTART:20260716T133000Z
DTEND:20260716T135500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/44
DESCRIPTION:Title: Sets with few subset sums\nby Noah Kravitz (Oxford University\, UK)
as part of Combinatorial and additive number theory seminar (CANT 2026)\n\
nLecture held in Science Center in the CUNY Graduate Center (4th floor).\n
\nAbstract\nA classical result of Nathanson shows that every $n$-element s
et of positive reals has at least $\\binom{n+1}{2}+1$ distinct subset sums
\, with equality exactly for homogeneous arithmetic progressions. We estab
lish stability versions of this inverse theorem in two regimes. First\, fo
r any parameter $0 \\leq M \\leq n-4$\, we precisely characterize the $n$-
element sets of positive reals with at most $\\binom{n+1}{2}+1+M$ subset s
ums. Second\, for any constant $C$\, we provide a characterization\, sharp
up to constants\, of the $n$-element sets of positive reals with at most
$Cn^2$ distinct subset sums. Joint work with Ruben Carpenter and Colin Def
ant.\n
LOCATION:https://researchseminars.org/talk/CANT2026/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan McNew (Towson University)
DTSTART:20260716T140000Z
DTEND:20260716T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/45
DESCRIPTION:Title: Matchable numbers\nby Nathan McNew (Towson University) as part of Co
mbinatorial and additive number theory seminar (CANT 2026)\n\nLecture held
in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nW
e say a natural number is matchable if there is a bijection from the set o
f $\\tau(n)$ divisors of $n$ to the set $[1\,2\,\\ldots\,\\tau(n)]$\, wher
e corresponding numbers are relatively prime. We show that the set of matc
hable numbers has an asymptotic density\, which we compute\, and we show t
hat every squarefree number is matchable. We also present some related uns
olved problems. This is joint work with Carl Pomerance.\n
LOCATION:https://researchseminars.org/talk/CANT2026/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Rényi Institute of Mathematics and Corvinus Univers
ity\, Hungary)
DTSTART:20260716T143000Z
DTEND:20260716T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/46
DESCRIPTION:Title: Lower bounds for mask polynomials with many cyclotomic divisors\nby
Gergely Kiss (Rényi Institute of Mathematics and Corvinus University\, Hu
ngary) as part of Combinatorial and additive number theory seminar (CANT 2
026)\n\nLecture held in Science Center in the CUNY Graduate Center (4th fl
oor).\n\nAbstract\nWe study finite subsets and multisets of cyclic groups
\\(\\mathbb{Z}_M\\)\nwhose mask polynomials have prescribed cyclotomic div
isors. More precisely\,\nif \\(A\\subseteq \\mathbb{Z}_M\\)\, we consider
its mask polynomial $$\n A(X)=\\sum_{a\\in A} X^a\n \\qquad \\text{in } \\
mathbb{Z}[X]/(X^M-1)\,\n$$ and ask how divisibility by selected cyclotomic
polynomials constrains\nthe size and structure of \\(A\\). \nThis questio
n is motivated by its connections with translational tilings\,\nthe Coven-
-Meyerowitz conjecture\, and one-dimensional Fuglede-type problems.\nWe pr
ove new lower bounds for the cardinality of such sets and develop several\
nstructural tools\, including \\\\\n & a truncation method and a multiscal
e extension of\nthe de Bruijn--Rédei--Schoenberg theorem. These results s
how that the\nexpected fibre-type extremal configurations do not always gi
ve the correct\nminimum once the prescribed cyclotomic divisors become suf
ficiently complicated. \nAt the same time\, in the two-dimensional case an
d in several further special\nsituations\, the lower bounds agree with the
natural fibre constructions. This is joint work with I. Łaba\, C. Marsha
ll\, and G. Somlai.\n
LOCATION:https://researchseminars.org/talk/CANT2026/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pramana Saldin (University of California\, Berkeley)
DTSTART:20260716T150000Z
DTEND:20260716T152500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/47
DESCRIPTION:Title: Left and right quotient sets in non-abelian groups\nby Pramana Saldi
n (University of California\, Berkeley) as part of Combinatorial and addit
ive number theory seminar (CANT 2026)\n\nLecture held in Science Center in
the CUNY Graduate Center (4th floor).\n\nAbstract\nFor a group $G$\, we d
efine the right quotient set and the left quotient set as follows: $$\n AA
^{-1}:=\\{a_1a_2^{-1}:a_1\,a_2\\in A\\} \\qquad A^{-1}A:=\\{a_1^{-1}a_2:a_
1\,a_2\\in A\\}.$$ \nWe examine the relationships between the left and rig
ht quotient sets. If $G$ is an abelian group\, then these sets are equal\,
but subtleties arise in non-abelian settings\, as these sets may not have
the same cardinality. Tao remarked that the cardinality difference $|AA^{
-1}| - |A^{-1}A|$ may be arbitrarily large for certain groups. \n\nWe firs
t give explicit constructions of sets $A$ where this difference attains ev
ery possible integer\, proving that the difference can be any possible val
ue if $G$ has elements of order 2. \n\nWe also find the minimum cardinalit
y of $A$ so that the difference between the cardinalities of the left and
right quotient sets is nonzero\, depending on the existence of order $2$ e
lements in $G$. \n\nTo prove these results\, we construct a graph called t
he difference graph $D_A$ that encodes equality in the right quotient set.
Similarly\, $D_{A^{-1}}$ encodes equality in the left quotient set. By ob
serving an isomorphism of edges in $D_A$ and $D_{A^{-1}}$ and counting con
nected components\, we are able to prove the results above. In the free gr
oup on two generators\, we can prove that the difference $|AA^{-1}| - |A^{
-1}A|$ is always even. We explicitly construct subsets of $F_2$ that achie
ve every even integer. In the infinite dihedral group $D_\\infty \\cong \\
mathbb{Z} \\rtimes \\mathbb{Z}/2$\, we prove that every integer difference
is achievable\, using the results of Martin and O'Bryant on the cardinali
ty differences of sum sets and difference sets in $\\mathbb{Z}.$ \n\nJoint
work with June Duvivier\, Xiaoyao Huang\, Ava Kennon\, Say-yeon Kwon\, St
even J. Miller\, Arman Rysmakhanov\, and Ren Watson\n
LOCATION:https://researchseminars.org/talk/CANT2026/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonah Klein (University of South Carolina)
DTSTART:20260716T153000Z
DTEND:20260716T155500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/48
DESCRIPTION:Title: The distortion method and its applications\nby Jonah Klein (Universi
ty of South Carolina) as part of Combinatorial and additive number theory
seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate
Center (4th floor).\n\nAbstract\nA covering system is a finite set of ari
thmetic progressions\, with the property that every integer belongs to at
least one of them. Covering systems were introduced by Erdös in 1950. In
the same article where he introduced them\, he asked if there was a unifor
m upper bound on the smallest modulus of covering systems with distinct mo
duli. This problem was resolved by Hough in 2015\, showing that the smalle
st modulus is always smaller than $10^{16}$. Expanding upon his work\, Bal
ister\, Bollobás\, Morris\, Sahasrabudhe\, and Tiba reduced this bound to
$616 000$\, with a method that they coined the distortion method. The aim
of this talk is to give a brief overview of the distortion method and its
applications\, with a particular focus on showing that it is impossible t
o construct 10 disjoint distinct covering systems. This is work in progres
s with Michael Filaseta and Alexandros Kalogirou.\n
LOCATION:https://researchseminars.org/talk/CANT2026/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Norbert Hegyvári (Eötvös University and Rényi Institute\, Hung
ary)
DTSTART:20260716T160000Z
DTEND:20260716T162500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/49
DESCRIPTION:Title: Consecutive sums\, Sidon sets\, convex sequences: Old and new problems\nby Norbert Hegyvári (Eötvös University and Rényi Institute\, Hunga
ry) as part of Combinatorial and additive number theory seminar (CANT 2026
)\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor
).\n\nAbstract\nIn the field of additive combinatorics\, sum-difference se
ts and subset\nsums have been extensively investigated.\nHowever\, the pro
perties and behavior of the so-called consecutive sums\nof sequences repre
sent a significantly less explored area.\nMy talk addresses this gap by di
scussing both old results and recent\ndevelopments concerning consecutive
sums\, highlighting their connections\nto Sidon sequences and convex seque
nces. Finally\, we will conclude the\ntalk by outlining several open quest
ions and problems.\n
LOCATION:https://researchseminars.org/talk/CANT2026/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amanda Montejano (Mexico)
DTSTART:20260716T173000Z
DTEND:20260716T175500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/50
DESCRIPTION:Title: Discrete Brunn–Minkowski inequalities\nby Amanda Montejano (Mexico
) as part of Combinatorial and additive number theory seminar (CANT 2026)\
n\nLecture held in Science Center in the CUNY Graduate Center (4th floor).
\n\nAbstract\nThe Brunn–Minkowski inequality is a cornerstone of convex
geometry\, with deep connections to several areas of mathematics. In recen
t years\, there has been growing interest in developing discrete versions
of this inequality. Attempts to formulate a discrete version of the Brunn
–Minkowski inequality naturally lead to problems in additive combinatori
cs\, particularly those involving lower bounds and structural aspects of f
inite sumsets in ${\\mathbb R}^d$ or ${\\mathbb Z}^d$. In the continuous s
etting\, a refinement due to Bonnesen incorporates the $(d-1)$-dimensional
volume of projections onto a hyperplane\, yielding sharper bounds that ca
pture geometric structure. A discrete counterpart of this refinement is cu
rrently known only in dimension two\, due to Grynkiewicz and Serra. In thi
s paper\, we explore extensions of this result to higher dimensions. In pa
rticular\, we introduce a framework for deriving discrete Brunn–Minkowsk
i-type inequalities in arbitrary dimension that incorporate projection dat
a of the underlying sets. This is a joint work with Oriol Serra and Luis M
ontejano.\n
LOCATION:https://researchseminars.org/talk/CANT2026/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isaac Rajagopal (MIT)
DTSTART:20260716T180000Z
DTEND:20260716T182500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/51
DESCRIPTION:Title: Possible sizes of sumsets\nby Isaac Rajagopal (MIT) as part of Combi
natorial and additive number theory seminar (CANT 2026)\n\nLecture held in
Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nNath
anson introduced the range of cardinalities of $h$-fold sumsets $ \\mathca
l{R}(h\,k):= \\{|hA|:A \\subseteq \\mathbb{Z} \\text{ and }|A| = k\\}. $ F
ollowing a remark of Erdös and Szemerédi that determined the form of $\\
mathcal{R}(h\,k)$ when $h=2$\, Nathanson asked what the form of $\\mathcal
{R}(h\,k)$ is for arbitrary $h\, k \\in \\mathbb{N}$. For $h \\in \\mathbb
{N}$\, we prove there is some constant $k_h \\in \\mathbb{N}$ such that if
$k > k_h$\, then $\\mathcal{R}(h\,k)$ is the entire interval $\\left[hk-h
+1\,\\binom{h+k-1}{h}\\right]$ except for a specified set of $\\binom{h-1}
{2}$ numbers. Moreover\, we show that one can take $k_3 = 2$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cosmin Pohoata (Emory University)
DTSTART:20260716T183000Z
DTEND:20260716T185500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/52
DESCRIPTION:Title: Sidon sets in the squares\, repeated distances\, and the Elekes-Ronyai p
roblem\nby Cosmin Pohoata (Emory University) as part of Combinatorial
and additive number theory seminar (CANT 2026)\n\nLecture held in Science
Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nWe discuss a
new combinatorial large-sieve method that uses algebraic splitting modulo
many small primes to turn local congruence restrictions into global constr
aints on repeated values. This has various applications\, for example: (i)
every Sidon subset of $\\{1^2\, 2^2\, \\ldots\, N^2\\}$ has size at most
$N \\cdot \\exp(-c \\log N / \\log \\log N)$\, the first super-polylogarit
hmic saving for a classical problem of Alon and Erdös\; (ii) a new upper
bound on the largest subset of $[N]^2$ with no repeated distances\, a prob
lem of Erdös and Guy\; and (iii) a new upper bound on the largest subset
of $[N]^2$ with no isosceles triangle\, a problem recently popularized by
Charton\, Ellenberg\, Wagner\, and Williamson. Based on recent joint work
with Ernie Croot\, Junzhe Mao\, Adam Sheffer\, and Kyle Yip. We will also
discuss how these ideas recently led to a counterexample for the Elekes--R
\\'onyai problem (and to a few other constructions).\n
LOCATION:https://researchseminars.org/talk/CANT2026/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amita Malik (Pennsylvania State University)
DTSTART:20260716T190000Z
DTEND:20260716T192500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/53
DESCRIPTION:Title: Lehmer-type partition statistics\nby Amita Malik (Pennsylvania State
University) as part of Combinatorial and additive number theory seminar (
CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center (
4th floor).\n\nAbstract\nMotivated by Lehmer’s work on weighted partitio
ns\, we discuss generalized (super)norms of various classes of partitions\
nand overpartitions. This is joint work with A. Dhar.\n
LOCATION:https://researchseminars.org/talk/CANT2026/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sandra Kingan (Brooklyn College and the Graduate Center\, CUNY)
DTSTART:20260716T193000Z
DTEND:20260716T195500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/54
DESCRIPTION:Title: Deletable edges in 3-connected graphs and their applications\nby San
dra Kingan (Brooklyn College and the Graduate Center\, CUNY) as part of Co
mbinatorial and additive number theory seminar (CANT 2026)\n\nLecture held
in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nI
will analyze 3-connected graphs that contain a fixed 3-connected graph $H
$ as a minor\, but in which no edge can be deleted while preserving 3-conn
ectivity and an $H$-minor. Let $G$ and $H$ be simple 3-connected graph su
ch that $G$ has an $H$-minor. An edge $e$ in $G$ is called $H$-deletabl
e if $G\\backslash e$ is 3-connected and has an $H$-minor. If $G$ has no $
H$-deletable edge\, then $G$ can be reduced to $H$ using three specific lo
cal operations. This gives a framework for studying extremal graphs with
no $H$-deletable edges and yields applications to excluded-minor question
s. This talk is based on a paper in Discrete Mathematics (Vol 349\, Issue
6).\n
LOCATION:https://researchseminars.org/talk/CANT2026/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:C. J. Mozzochi (Connecticut)
DTSTART:20260716T200000Z
DTEND:20260716T202500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/55
DESCRIPTION:Title: A new approach to the circle method attack on the m-prime conjecture
\nby C. J. Mozzochi (Connecticut) as part of Combinatorial and additive nu
mber theory seminar (CANT 2026)\n\nLecture held in Science Center in the C
UNY Graduate Center (4th floor).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O’Bryant (College of Staten Island and CUNY Graduate Cente
r)
DTSTART:20260716T203000Z
DTEND:20260716T205500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/56
DESCRIPTION:Title: Problem Session\nby Kevin O’Bryant (College of Staten Island and C
UNY Graduate Center) as part of Combinatorial and additive number theory s
eminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate
Center (4th floor).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaurav Kumar (Indian Institute of Technology Gandhinagar\, India)
DTSTART:20260717T120000Z
DTEND:20260717T122500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/57
DESCRIPTION:Title: Some identities of the sums-of-tails type\nby Gaurav Kumar (Indian I
nstitute of Technology Gandhinagar\, India) as part of Combinatorial and a
dditive number theory seminar (CANT 2026)\n\nLecture held in Science Cente
r in the CUNY Graduate Center (4th floor).\n\nAbstract\nA new sums-of-tail
s identity involving two parameters b and d is obtained and is used to der
ive more results of similar type. One of Ramanujan’s sums-of-tails ident
ities from the Lost Notebook is shown to be a special case of our result.
In the course of deriving Ramanujan’s identity\, we obtain a new result
of combinatorial significance. Two new representations for an infinite ser
ies associated to a mock theta function are derived. Also\, we give an app
lication of an identity of Andrews and Onofri. This talk is based on joint
work with Prof. Atul Dixit and Aviral Srivastava.\n
LOCATION:https://researchseminars.org/talk/CANT2026/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivekanand Goswami (Indian Institute of Technology Gandhinagar\, I
ndia)
DTSTART:20260717T123000Z
DTEND:20260717T125500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/58
DESCRIPTION:Title: Restricted set addition in finite abelian groups\nby Vivekanand Gosw
ami (Indian Institute of Technology Gandhinagar\, India) as part of Combin
atorial and additive number theory seminar (CANT 2026)\n\nLecture held in
Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $
A$ be a nonempty subset of a finite abelian group $G$ of order $n$. For an
integer $h \\geq 2$\, the restricted $h$-fold sumset $h^\\wedge A$ is the
set of all sums of $h$ distinct elements of $A$. It is known that if $G$
is a group of order $n$ and $A$ is a subset of $G$ such that $|A| > \\frac
{n}{2}$\, then $h^{\\wedge}A = G$ under some conditions on $h$ and $n$. Wh
ile the constant $1/2$ is optimal for groups of even order\, it is not opt
imal for groups of odd order. For an integer $h \\geq 4$\, let $\\alpha_h$
be the unique positive root of the polynomial $3^{h - 2} x^{h - 1} + x -
1$. In this talk\, we discuss that for any $\\alpha > \\alpha_h$\, there e
xists a positive integer $M_h(\\alpha)$\, which is determined precisely\,
such that for all $n > M_h(\\alpha)$ with $n$ odd\, if $A$ is a subset of
a finite abelian group $G$ of order $n$ and if $|A| \\geq \\alpha n$\, the
n $h^{\\wedge} A = G$. Moreover\, $\\alpha_h > \\alpha_{h + 1}$ for $h \\g
eq 4$ and $\\alpha_h$ approaches $\\frac{1}{3}$ as $h$ increases\, and the
constant $\\frac{1}{3}$ is optimal when the smallest prime dividing $n$ i
s $3$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Semin Yoo (Institute for Basic Science\, Korea)
DTSTART:20260717T130000Z
DTEND:20260717T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/59
DESCRIPTION:Title: Multiplicative irreducibility of shifted multiplicative subgroups\nb
y Semin Yoo (Institute for Basic Science\, Korea) as part of Combinatorial
and additive number theory seminar (CANT 2026)\n\nLecture held in Science
Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nA central th
eme in additive combinatorics is the interplay between addition and multip
lication. Roughly speaking\, sets with strong multiplicative structure are
not expected to exhibit rich additive structure\, and vice versa. In a re
cent breakthrough\, Kalmynin resolved a conjecture of Lev--Sonn and Sárk
özy on additive decompositions of multiplicative subgroups in prime field
s and quadratic residues. Motivated by this work\, we study multiplicative
analogues of these questions. We show that\, under a certain additional c
ondition\, a shifted multiplicative subgroup cannot be written as a produc
t set\, and that it also cannot be written as a ratio set unconditionally.
In this talk\, I will discuss these results and the main ideas of the pro
ofs. This talk is based on joint work with Seoyoung Kim (University of Bas
el) and Chi Hoi Yip (Georgia Institute of Technology).\n
LOCATION:https://researchseminars.org/talk/CANT2026/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sándor Kiss (Budapest University of Technology and Economics\, Hu
ngary)
DTSTART:20260717T133000Z
DTEND:20260717T135500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/60
DESCRIPTION:Title: Monotone increasing representation functions\nby Sándor Kiss (Budap
est University of Technology and Economics\, Hungary) as part of Combinato
rial and additive number theory seminar (CANT 2026)\n\nLecture held in Sci
ence Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $k\\
ge 2$ be an integer and let $A$ be a set of nonnegative integers. The repr
esentation function $R_{A\,k}(n)$ for the set $A$ is the number of represe
ntations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. A
few years ago\, Bell and Shallit constructed a set $A$ of natural numbers
such that $\\mathbb{N}\\setminus A$ is infinite\, but the corresponding re
presentation function is strictly increasing. Later\, together with Csaba
Sándor and Yang Quan-Hui\, we improved their result. Furthermore\, we con
structed a dense set such that the corresponding representation function i
s not strictly increasing. In my talk I will also give an overview of the
recent progress on this topic.\n
LOCATION:https://researchseminars.org/talk/CANT2026/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dennis Eichhorn (University of California - Irvine)
DTSTART:20260717T140000Z
DTEND:20260717T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/61
DESCRIPTION:Title: The combinatorics of sequences that enjoy a curious self-convolutive pro
perty\nby Dennis Eichhorn (University of California - Irvine) as part
of Combinatorial and additive number theory seminar (CANT 2026)\n\nLecture
held in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstra
ct\nIn 2002\, Andrews\, Lewis\, and Lovejoy introduced the combinatorial o
bjects called partitions with designated summands.\nIf we restrict our att
ention to $\\mathrm{PDO}(n)$\, the number of partitions with designated su
mmands in which all parts are odd\, a very curious property emerges.\nThe
very unexpected identity $\\qquad\n \\sum_{n=0}^\\infty \\mathrm{PDO}(2n)q
^n = \\left ( \\sum_{n=0}^\\infty \\mathrm{PDO}(n)q^n \\right )^2\n$ holds
.\nThat is\, the sequence $\\{\\mathrm{PDO}(2n)\\}_{n=0}^\\infty$ is the c
onvolution of the sequence $\\{\\mathrm{PDO}(n) \\}_{n=0}^\\infty$ with it
self!\nSequences sharing this curious property are now called ``$2$-convol
utive\,'' and a small handful of such sequences appear in the OEIS. Many a
uthors have called for a combinatorial proof of the $2$-convolutivity of $
\\mathrm{PDO}(n)$. After a nearly two-year-long collaboration with Chern\,
Fu\, and Sellers\, we are happy to announce that we have finally found th
e requested combinatorial proof.\nIn this talk\, we discuss this new proof
\, along with the combinatorial proofs of the $2$-convolutivity of several
other partition functions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kalmynim (Higher School of Economics\, Russia)
DTSTART:20260717T143000Z
DTEND:20260717T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/62
DESCRIPTION:Title: Sárközy’s conjecture on quadratic residues\nby Alexander Kalmyni
m (Higher School of Economics\, Russia) as part of Combinatorial and addit
ive number theory seminar (CANT 2026)\n\nLecture held in Science Center in
the CUNY Graduate Center (4th floor).\n\nAbstract\nFor an odd prime numbe
r $p$\, let $\\mathcal R_p\\subset \\mathbb F_p$ be the set of all non-zer
o quadratic residues. A. Sárközy conjectured that the set $\\mathcal R_p
$ does not admit a non-trivial additive decomposition for large enough $p$
\, i.e. for $p>p_0$ the identity $A+B=\\mathcal R_p$ implies $\\min(|A|\,|
B|)=1$. In this talk we present a complete resolution of Sárközy's conje
cture. Further\, we show that\, for a subgroup $G\\subset \\mathbb F_p^*$\
, the equality $G\\cup\\{0\\}=A-A$ for some $A$ implies $|G|=2$ or $6$ and
if $G=A+B$ non-trivially\, then $|A|=|B|=\\sqrt{|G|}$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohan (BK Birla Institute of Engineering and Technology\, Pilani\,
India)
DTSTART:20260717T150000Z
DTEND:20260717T152500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/63
DESCRIPTION:Title: Lehmer-type conjectures and open problems for Nathanson’s totient func
tions\nby Mohan (BK Birla Institute of Engineering and Technology\, Pi
lani\, India) as part of Combinatorial and additive number theory seminar
(CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center
(4th floor).\n\nAbstract\nNathanson’s totient functions $\\Phi(n)$ and $
\\Phi_k(n)$\, where $\\Phi(n)$ counts the number of nonempty sets $A \\sub
seteq \\{1\, 2\, \\dots\, n\\}$ for which $\\gcd(A)$ is relatively prime t
o $n$\, and $\\Phi_k(n)$ restricts those of size $k$. We formulate and ana
lyze some analogue of Lehmer's conjecture in the setting of Nathanson’s
totient functions $\\Phi(n)$ and $\\Phi_k(n)$. We further discuss divisibi
lity phenomena for $\\Phi(n)$. We conclude with several conjectures and op
en problems concerning density\, arithmetic progressions\, and further str
uctural properties of these functions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasuaki Gyoda (Nagoya University\, Japan)
DTSTART:20260717T153000Z
DTEND:20260717T155500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/64
DESCRIPTION:Title: Generalized discrete Lagrange–Markov spectra\nby Yasuaki Gyoda (Na
goya University\, Japan) as part of Combinatorial and additive number theo
ry seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Gradu
ate Center (4th floor).\n\nAbstract\nThis talk concerns a discrete extensi
on of the classical Lagrange and Markov\nspectra\, motivated by generalize
d Markov equations. In the classical case\,\nthe discrete spectral values
below $3$ are organized by Markov numbers and are\ndescribed through conti
nued fractions\, Christoffel words\, and Cohn matrices.\nI will explain ho
w an analogous picture can be developed for generalized\nMarkov numbers ar
ising from \n $$ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy\n =(3+k_1+k_2+k_3)xyz.\n$$\
nFor each generalized Markov number\, one obtains an explicit spectral val
ue\nwhich is realized both as the Lagrange constant of a quadratic irratio
nal and\nas the Markov constant of an indefinite binary quadratic form wit
h rational\ncoefficients. The emphasis of the talk will be on the main ide
a of the\nconstruction: generalized Cohn matrices and symbolic sequences c
oming from\nstraight-line codings play the role classically played by Chri
stoffel words and \nCohn matrices. The aim is to present a combinatorial a
nd matrix-theoretic\nframework for viewing classical and generalized discr
ete Diophantine spectra in\na unified way.\n
LOCATION:https://researchseminars.org/talk/CANT2026/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philippa Holdridge (Alfréd Rényi Institute\, Hungary)
DTSTART:20260717T160000Z
DTEND:20260717T162500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/65
DESCRIPTION:Title: The size of certain symmetric differences of sets of integers\nby Ph
ilippa Holdridge (Alfréd Rényi Institute\, Hungary) as part of Combinato
rial and additive number theory seminar (CANT 2026)\n\nLecture held in Sci
ence Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nConsider
a set $A\\subseteq \\mathbb{N}$ which is finite and nonempty. Letting $\\
Delta$ denote the symmetric difference of sets and $k\\cdot A=\\{ka:a\\in
A\\}$\, it can be shown that $A\\Delta (2\\cdot A)$ always contains at lea
st two elements. It also turns out that $A\\Delta (2\\cdot A) \\Delta (3\\
cdot A)$ has at least three elements. Does $A\\Delta (2\\cdot A)\\Delta\\c
dots \\Delta (n\\cdot A)$ have at least $n$ elements for all $n\\in \\math
bb{N}$? This question was posed by Pilz in an equivalent form involving th
e minimal distance of certain linear codes. If true\, then this lower boun
d is best possible\, as seen by considering $A=\\{1\\}$. The lower bound i
s also attained when $A=\\{1\,2\,\\dots\,n\\}$ and\, in fact\, for each $n
$\, there are arbitrarily large sets $A$ such that $A\\Delta (2\\cdot A)\\
Delta\\cdots \\Delta (n\\cdot A)$ has exactly $n$ elements.\n\nPilz proved
the conjecture for $n\\le 6$\, and it can also be proven for $n=7$ and $8
$. For larger $n$\, Pach and Szabó proved a lower bound of the form $n/(\
\log n)^{\\lambda}$ for $\\lambda\\approx 0.22$. Until recently\, this was
the strongest result known\, but in a recent work\, we have proven the co
njecture for all sufficiently large $n$. \nMore precisely\, whenever $n\\g
e 3^{81}$. In this talk we will outline the proof and discuss some related
problems. Joint work with P. Pach.\n
LOCATION:https://researchseminars.org/talk/CANT2026/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Leonetti (Universit`a degli Studi dell’Insubria and Univer
sit`a Bocconi\, Italy)
DTSTART:20260717T163000Z
DTEND:20260717T165500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/66
DESCRIPTION:Title: Independent families and asymptotic density\nby Paolo Leonetti (Univ
ersit`a degli Studi dell’Insubria and Universit`a Bocconi\, Italy) as pa
rt of Combinatorial and additive number theory seminar (CANT 2026)\n\nLect
ure held in Science Center in the CUNY Graduate Center (4th floor).\n\nAbs
tract\nLet $\\mathcal{D}$ be the family of sets $S\\subseteq \\mathbb{N}$
for which the asymptotic density $$\nd(S):=\\lim_{n\\to \\infty}\\frac{|S\
\cap [1\,n]|}{n}\n$$\nexists. Treating $d$ as a finitely additive probabil
ity measure on $\\mathcal{D}$\, we study structural properties of families
of sets $\\mathcal{A}\\subseteq \\mathcal{P}(\\mathbb{N})$ which are inde
pendent (in its classical statistical meaning). We conclude with several o
pen questions. Reference: \nJ. Keith and P. Leonetti\, On maximal families
of independent sets with respect to asymptotic density \, https://arxiv.o
rg/abs/2603.28922.\n
LOCATION:https://researchseminars.org/talk/CANT2026/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neetu (National Institute of Technology Karnataka\, India)
DTSTART:20260717T170000Z
DTEND:20260717T172500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/67
DESCRIPTION:Title: Small doubling in right-ordered groups\nby Neetu (National Institute
of Technology Karnataka\, India) as part of Combinatorial and additive nu
mber theory seminar (CANT 2026)\n\nLecture held in Science Center in the C
UNY Graduate Center (4th floor).\n\nAbstract\nFreiman conjectured that if
$S$ is a finite subset of a torsion-free group $G$ with $k\\geq 3$ element
s and $|S^{2}|\\leq 3k-4\,$ then $S$ is a subset of a small geometric prog
ression of length at most $2k-3$. In 2014\, Freiman et al. settled this co
njecture when $S$ is a finite subset of an ordered group. In this talk\, w
e study this problem in the broader framework of right-ordered groups. Und
er suitable structural conditions on the subset $S$\, we discuss results t
hat extend aspects of Freiman's conjecture to this setting. We further foc
us on the right-ordered Baumslag--Solitar group $$\\text{BS}(1\,q) = \\lan
gle a\, b \\mid ab = b^q a \\rangle\, \\quad q \\in \\mathbb{Z}.$$ We show
that for $q \\neq -1$\, if $S$ is a finite subset of $\\text{BS}(1\,q)$ w
ith the identity element as its minimum and satisfying $|S^2| \\leq 3|S| -
4$\, then the subgroup generated by $S$ is abelian. This is joint work wi
th Mohan and B. R. Shankar. The results are based on our recent paper: htt
ps://link.springer.com/article/10.1007/s00025-025-02576-2.\n
LOCATION:https://researchseminars.org/talk/CANT2026/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Veronica Bitonti (University of Oxford\, UK)
DTSTART:20260717T173000Z
DTEND:20260717T175500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/68
DESCRIPTION:Title: Gap sets of random generalized numerical semigroups\nby Veronica Bit
onti (University of Oxford\, UK) as part of Combinatorial and additive num
ber theory seminar (CANT 2026)\n\nLecture held in Science Center in the CU
NY Graduate Center (4th floor).\n\nAbstract\nFor a fixed positive integer
$d$ and a small real $p>0$\, sample a $p$-random subset $A \\subseteq \\ma
thbb{Z}_{\\geq 0}^d$\, and let $S:=\\langle A \\rangle$ be the generalized
numerical semigroup generated by $A$. We show that\, with high probabilit
y (as $p \\to 0$)\, the gap set $\\mathbb{Z}_{\\geq 0}^d \\setminus S$ is
well approximated by the shifted hyperboloid region $$\\{(x_1\, \\ldots\,
x_d) \\in \\mathbb{R}_{\\geq 0}^d: (x_1+\\log p^{-1}) \\cdots (x_d+\\log p
^{-1})\\ll p^{-1}(\\log p^{-1})^{d+1}\\}.$$ This generalizes work of Kravi
tz\, Morales\, and Schildkraut on the $1$-dimensional setting. We also obt
ain the same result with $S$ replaced by the set of subset sums of $A$. Th
is is a joint work with Noah Kravitz.\n
LOCATION:https://researchseminars.org/talk/CANT2026/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Salvatore Tringali (Hebei Normal University\, China)
DTSTART:20260717T180000Z
DTEND:20260717T182500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/69
DESCRIPTION:Title: Power semigroups and two rigidity theorems for groups\nby Salvatore
Tringali (Hebei Normal University\, China) as part of Combinatorial and ad
ditive number theory seminar (CANT 2026)\n\nLecture held in Science Center
in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $\\mathcal P(H)
$ be the semigroup obtained by endowing the family of all non-empty subset
s of a semigroup $H$ with the setwise operation naturally induced by $H$ o
n its power set\, and denote by $\\mathcal P_\\text{fin}(H)$ the subsemigr
oup of $\\mathcal P(H)$ consisting of all non-empty finite subsets of $H$.
We call $\\mathcal P(H)$ and $\\mathcal P_\\text{fin}(H)$ the large power
semigroup and the finitary power semigroup of $H$\, respectively.\n\nWe s
how that if $H$ is a group and $K$ is an arbitrary semigroup\, then for\n$
\\mathcal P(H)$ to be isomorphic to $\\mathcal P(K)$ it is necessary (and
sufficient) that $H$ is isomorphic to $K$ (and hence $K$ is itself a group
). The finitary\nanalogue of the same statement appears to be considerably
more difficult\,\nand we establish it only when $H$ is an additive subgro
up of the\nrationals. The proof of this second result\nrelies\, in a circu
itous way\, on a special case of the Evertse--Schlickewei--Schmidt\ntheore
m. The talk is based on joint work with Shuolin Liu.\n
LOCATION:https://researchseminars.org/talk/CANT2026/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katalin Gyarmati (Eötvös Loránd University\, Hungary)
DTSTART:20260717T183000Z
DTEND:20260717T185500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/70
DESCRIPTION:Title: The taxicab problem for polynomials and generalizations of Mason’s the
orem\nby Katalin Gyarmati (Eötvös Loránd University\, Hungary) as p
art of Combinatorial and additive number theory seminar (CANT 2026)\n\nLec
ture held in Science Center in the CUNY Graduate Center (4th floor).\n\nAb
stract\nThis talk is motivated by Ramanujan's famous taxicab problem and i
s concerned with the solvability of polynomial equations of the form $p^n+
q^n=r^n+s^n$ and\, more generally\, $p_1^{k_1}+\\dots+p_m^{k_m}=0$ over th
e complex numbers. Using Wronskian determinants and Mason's theorem\, we o
btain sharp upper bounds for the exponents. In particular\, we will show t
hat there are no relatively prime polynomials (with at least one non-const
ant) satisfying the generalised taxicab equation for $n \\ge 16$. We also
consider an extension of Mason's theorem to $f_0+f_1+\\dots+f_k=0$ for sev
eral polynomial terms over the complex numbers and finite fields\, obtaini
ng the corresponding degree bounds. Finally\, the talk points out interes
ting future cryptographic applications of these theoretical results\, in p
articular\, the construction of large families of pseudorandom binary sequ
ences with small cross-correlation measures.\n
LOCATION:https://researchseminars.org/talk/CANT2026/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Hendel (Towson University)
DTSTART:20260717T190000Z
DTEND:20260717T192500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/71
DESCRIPTION:Title: Improvements in calculating the recursion satisfied by a family of deter
minants\nby Russell Hendel (Towson University) as part of Combinatoria
l and additive number theory seminar (CANT 2026)\n\nLecture held in Scienc
e Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nA variety o
f problems can be elegantly solved by identifying the recursion satisfied
by the determinants of a family of matrices. In 2016\, Jia\, Yang\, and Li
provided a general 6-th order recursion for the family of arbitrary pentd
iagonal Toeplitz matrices by using Laplace expansions. Recently\, Evans an
d Hendel showed that this method is potentially generalizable and applied
it to prove an outstanding conjecture on resistance distance in linear 3-t
rees. However\, Evans and Hendel left as an open problem the convergence o
f their procedure in the general case. Hendel has recently proven converge
nce for such a Laplace-expansion approach for an arbitrary family of squar
e\, banded\, Toeplitz matrices with $k$ super and sub diagonals for any po
sitive integer $k.$ Hendel also eliminated the computational matrix method
s of Evans and Hendel replacing them with a simpler algebraic manipulative
system. This note supplements this procedure by showing an improved metho
d to solve the resulting system of several simultaneous equations in famil
ies of determinants. This improved procedure\, applied to explore an outst
anding conjecture of Bareett\, Evans\, and Francis on the general $k$-line
ar tree\, uncovers several interesting patterns which are presented.\n
LOCATION:https://researchseminars.org/talk/CANT2026/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjil P. Saikia (Ahmedabad University\, Ahmedabad\, India)
DTSTART:20260717T193000Z
DTEND:20260717T195500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/72
DESCRIPTION:Title: Hook length biases in t-regular and t-core partitions\nby Manjil P.
Saikia (Ahmedabad University\, Ahmedabad\, India) as part of Combinatorial
and additive number theory seminar (CANT 2026)\n\nLecture held in Science
Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nRecently\, t
he theory of hook length biases has emerged as a prominent research topic.
Led by Ballantine\, Burson\, Craig\, Folsom\, and Wen\, hook length biase
s are being explored for ordinary partitions\, odd versus distinct partiti
ons\, self-conjugate versus distinct odd partitions. Recently\, Singh and
Barman opened the door to hook length biases in $t$-regular partitions as
well. \n\nThe objective of this talk is two fold. First\, we present a pre
viously unobserved connection of hook-lengths in $t$-regular partitions wi
th certain distinct parts partitions. Second\, we extend the theory of hoo
k length biases to $t$-core partitions. For example\, let $a_{t\,k}(n)$ de
note the number of hooks of length $k$ in all $t$-core partitions of $n$\,
then we find that $a_{3\,1}(n) \\ge a_{3\,2}(n) \\ge a_{3\,4}(n)$ and $a_
{4\,1}(n) \\ge a_{4\,3}(n)$ for all $n$. Most of the methods employed in t
his work are combinatorial. Joint work with Talukdar\; and Baruah\, Das\,
and Mahanta.\n
LOCATION:https://researchseminars.org/talk/CANT2026/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neranga Fernando (Knox College)
DTSTART:20260717T200000Z
DTEND:20260717T202500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/73
DESCRIPTION:Title: Contributions to the famiily of reversed Dickson polynomials\nby Ner
anga Fernando (Knox College) as part of Combinatorial and additive number
theory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY G
raduate Center (4th floor).\n\nAbstract\nLet $p$ be a prime\, $q$ a power
of $p$\, and $\\mathbb{F}_q$ the finite field with $q$ elements. A polynom
ial $f\\in \\mathbb{F}_q[\\tt X]$ is called a permutation polynomial of $\
\mathbb{F}_q$ if the associated mapping $\\tt X\\mapsto f(\\tt X)$ from $\
\mathbb{F}_q$ to $\\mathbb{F}_q$ is a permutation of $\\mathbb{F}_q$. Perm
utation polynomials have gained widespread attention due to their applicat
ions in cryptography\, coding theory\, and combinatorics. The $n$th revers
ed Dickson polynomial is given by the explicit expression \n$$ D_n(a\,\\tt
X)=\\sum_{i=0}^{\\lfloor n/2\\rfloor}\\\,\\frac{n}{n-i}\\\,\\binom{n-i}{i
}\\\,a^{n-2i}\\\,(-\\tt X)^i $$ where $a\\in \\mathbb{F}_q$ is a parameter
. Reversed Dickson polynomials have played an important role in the area o
f permutation polynomials since their introduction in 2009. \n\nA self-rec
iprocal polynomial is a polynomial whose coefficients form a palindrome. S
elf-reciprocal polynomials have important applications in coding theory. I
n this talk\, I will first speak about my contribution to the areas of per
mutation polynomials over finite fields and self-reciprocal polynomials vi
a reversed Dickson polynomials. I will also speak about a recent REU proje
ct conducted with my students at College of the Holy Cross on reversed Dic
kson permutation polynomials. Moreover\, I will present a list of research
projects for students.\n
LOCATION:https://researchseminars.org/talk/CANT2026/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lindsay Dever (Millersville University)
DTSTART:20260717T203000Z
DTEND:20260717T205500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/74
DESCRIPTION:Title: Atoms in the semigroup of non-negative integer matrices\nby Lindsay
Dever (Millersville University) as part of Combinatorial and additive numb
er theory seminar (CANT 2026)\n\nLecture held in Science Center in the CUN
Y Graduate Center (4th floor).\n\nAbstract\nIn the semigroup of $2\\times
2$ matrices with non-negative integer entries and non-zero determinant\, w
e study the factorization of matrices into atoms\, or irreducible matrices
. In 2022\, Baeth et al. discovered classes of atoms in this semigroup\; h
owever\, the factorability of most matrices remains unknown. As the result
of joint work with Eva Goedhart\, Gregory Heilbrunn\, and Tony W. H. Wong
\, I will discuss additional classes of atoms: a class of atoms with deter
minant $p$\, $2p$\, or $4p$\, where $p$ is prime\, and a class of atoms wh
ere the main diagonal is much ``larger'' than the off-diagonal (or vice-ve
rsa). In addition\, we find that bisymmetric matrices with relatively prim
e entries are a divisor-closed subset and use a factor-search algorithm to
classify bisymmetric atoms with minimum entry up to 4000.\n
LOCATION:https://researchseminars.org/talk/CANT2026/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasuaki Gyoda (Nagoya University\, Japan)
DTSTART:20260718T110000Z
DTEND:20260718T115500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/75
DESCRIPTION:Title: Generalized discrete Lagrange–Markov spectra\nby Yasuaki Gyoda (Na
goya University\, Japan) as part of Combinatorial and additive number theo
ry seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Gradu
ate Center (4th floor).\n\nAbstract\nThis talk concerns a discrete extensi
on of the classical Lagrange and Markov\nspectra\, motivated by generalize
d Markov equations. In the classical case\,\nthe discrete spectral values
below $3$ are organized by Markov numbers and are\ndescribed through conti
nued fractions\, Christoffel words\, and Cohn matrices.\nI will explain ho
w an analogous picture can be developed for generalized\nMarkov numbers ar
ising from \n $$ x^2+y^2+z^2+k_1yz+k_2zx+k_3xy\n =(3+k_1+k_2+k_3)xyz.\n$$\
nFor each generalized Markov number\, one obtains an explicit spectral val
ue\nwhich is realized both as the Lagrange constant of a quadratic irratio
nal and\nas the Markov constant of an indefinite binary quadratic form wit
h rational\ncoefficients. The emphasis of the talk will be on the main ide
a of the\nconstruction: generalized Cohn matrices and symbolic sequences c
oming from\nstraight-line codings play the role classically played by Chri
stoffel words and \nCohn matrices. The aim is to present a combinatorial a
nd matrix-theoretic\nframework for viewing classical and generalized discr
ete Diophantine spectra in\na unified way.\n
LOCATION:https://researchseminars.org/talk/CANT2026/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chahat Ahuja (Indraprastha Institute of Information Technology\, I
ndia)
DTSTART:20260718T120000Z
DTEND:20260718T122500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/76
DESCRIPTION:Title: Visibility of lattice points across polynomial curves\nby Chahat Ahu
ja (Indraprastha Institute of Information Technology\, India) as part of C
ombinatorial and additive number theory seminar (CANT 2026)\n\nLecture hel
d in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\n
The visibility of lattice points from the origin along a polynomial family
of curves constitutes a significant generalization of visibility along st
raight lines.\nFollowing the classical notion\, where the density of visib
le lattice points equals\n$1/\\zeta(2)$\, and its generalization to monomi
al curves of the form $y = ax^b$\,\nwhere the density equals $1/(b+1)$\, w
e study a family of polynomial curves defined\nby $$ \n y \\\;=\\\; q\\big
l(a_n x^n + a_{n-1}x^{n-1} + \\cdots + a_1 x\\bigr)\,\n$$ where $q$ is a p
ositive rational number.\n\nWe introduce a new criterion based on a \\emph
{polynomial greatest common divisor\ncondition} that provides a lower boun
d on the number of visible lattice points in\n$\\mathbb{N}^2$. Conversely\
, we derive conditions under which a given lattice point\nbecomes the next
visible point along such a polynomial curve. Using the\nprinciple of incl
usion-exclusion\, we obtain an exact double-sum formula for the\nnumber of
pairs $(a\, b) \\leq N$ that are visible with respect to this polynomial\
nfamily. \nFinally\, we extend the framework to related problems and pose
several open\nquestions concerning gap distributions and quantitative boun
ds for non-visible\npoints. This work provides a broader theoretical found
ation for lattice point\nvisibility beyond linear and monomial settings.\n
LOCATION:https://researchseminars.org/talk/CANT2026/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Baczkowski (University of Findlay\, Australia)
DTSTART:20260718T123000Z
DTEND:20260718T125500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/77
DESCRIPTION:Title: Building off the ideas of Erdós\, Sierpiński\, Riesel\, and more\n
by Daniel Baczkowski (University of Findlay\, Australia) as part of Combin
atorial and additive number theory seminar (CANT 2026)\n\nLecture held in
Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nIn 19
50\, Erd\\H{o}s proved there are infinitely many odd integers that are not
of the form $2^k + p$\, where $p$ is a prime. \nIn 1956\, using similar m
ethods\, Riesel proved there are infinitely many odd integers $k$ such tha
t $k\\cdot 2^n - 1$ is composite for all positive integers~$n$. Then\, in
1960\, Sierpi\\'{n}ski proved that there are infinitely many odd integers
$\\ell$ such that $\\ell\\cdot 2^n + 1$ is composite for all positive inte
gers $n$. \nWe will discuss various other related results such as how some
classical sequences like Fibonacci\, triangular\, and more intersect the
set of all possible Reisel and/or Sierpiński numbers.\n
LOCATION:https://researchseminars.org/talk/CANT2026/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eshita Mazumdar (Ahmedabad University\, Ahmedabad\, India)
DTSTART:20260718T130000Z
DTEND:20260718T132500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/78
DESCRIPTION:Title: Extending zero-sum theory from abelian to non-abelian groups\nby Esh
ita Mazumdar (Ahmedabad University\, Ahmedabad\, India) as part of Combina
torial and additive number theory seminar (CANT 2026)\n\nLecture held in S
cience Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nZero-s
um theory is a central topic in additive combinatorics that studies the st
ructure of sequences over finite groups and the conditions guaranteeing th
e existence of zero-sum subsequences. Fundamental parameters in this area
include the Davenport constant and the Erdős–Ginzburg–Ziv constant\,
which measure the threshold lengths forcing zero-sum behavior. These invar
iants originated in the study of non-unique factorizations in algebraic nu
mber theory\, but determining their exact values remains a challenging pro
blem even for many finite abelian groups. In this talk\, I will discuss re
cent progress on zero-sum problems in finite non-abelian groups. In partic
ular\, I will highlight how combinatorial techniques developed for abelian
groups can be adapted—or fail—to extend to the non-abelian setting\,
and how new phenomena arise due to the lack of commutativity. I will also
present several results that reveal surprising connections between classic
al zero-sum invariants of abelian groups and their analogues for non-abeli
an groups\, pointing toward a broader combinatorial framework for zero-sum
theory.\n
LOCATION:https://researchseminars.org/talk/CANT2026/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krystian Gajdzica (Jagiellonian University\, Poland)
DTSTART:20260718T133000Z
DTEND:20260718T135500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/79
DESCRIPTION:Title: On the Bessenrodt-Ono inequality for polynomials\nby Krystian Gajdzi
ca (Jagiellonian University\, Poland) as part of Combinatorial and additiv
e number theory seminar (CANT 2026)\n\nLecture held in Science Center in t
he CUNY Graduate Center (4th floor).\n\nAbstract\nIn 2016\, Bessenrodt and
Ono proved that the partition function satisfies the inequality $\n p(a)p
(b)>p(a+b)\n$\nfor all $a\,b\\geqslant2$ with $a+b>9$. Since then\, analog
ous properties have been investigated for many partition statistics. In th
is talk\, following Gian-Carlo Rota's advice\, we move from the discrete p
roblem to the continuous one\, and consider a family of recursively define
d polynomials \n$\nP_n^g(x) := \\frac{x}{n} \\sum_{k=1}^n g(k) P_{n-k}^g(x
)\n$\nwith the initial condition $P_0^g(x):=1$\, where $(g(n))_{n\\in\\mat
hbb{N}}$ is an arbitrary sequence of positive real numbers such that $g(1)
=1$. We derive an efficient criterion characterizing when the inequality \
n$\n P_{a}^g(x)P_{b}^g(x)\\geqslant P_{a+b}^g(x)\n$\n is satisfied for all
$x\\geqslant x_0$ and $a\,b\\geqslant1$\, where $x_0$ is some real number
depending on $g$. Moreover\, we illustrate the usefulness of this criter
ion by applying it to various combinatorial sequences.\n
LOCATION:https://researchseminars.org/talk/CANT2026/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debyani Manna (Indian Institute of Technology Roorkee\, India)
DTSTART:20260718T140000Z
DTEND:20260718T142500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/80
DESCRIPTION:Title: Extended Inverse results for restricted h-fold sumset in integer\nby
Debyani Manna (Indian Institute of Technology Roorkee\, India) as part of
Combinatorial and additive number theory seminar (CANT 2026)\n\nLecture h
eld in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract
\nLet $A$ be a finite set of $k$ integers. For $2 \\leq h \\leq k$\, the r
estricted h-fold sumset $h^{\\wedge}A$ is the set of all sums of $h$ disti
nct elements of the set $A$. In additive combinatorics\, much of the focus
has traditionally been on finite integer sets whose sumsets are unusually
small (cf. Freiman’s theorem and its extensions). More recently\, Natha
nson posed the inverse problem for the restricted sumset $h^{\\wedge}A$ wh
en $|h^{\\wedge}A|$ is small. For $h \\in \\{2\,3\,4\\}$\, this question h
as already been studied by Mohan and Pandey. In this article\, we study th
e inverse problems for $h^{\\wedge}A$ with arbitrary $h \\geq 3$ and chara
cterize all possible sets $A$ for certain cardinalities of $h^{\\wedge}A$.
Joint work with Mohan and Ram Krishna Pandey.\n
LOCATION:https://researchseminars.org/talk/CANT2026/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chi Hoi Yip (Hong Kong University of Science and Technology\, Hong
Kong)
DTSTART:20260718T143000Z
DTEND:20260718T145500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/81
DESCRIPTION:Title: Additive properties of multiplicatively defined sets\nby Chi Hoi Yip
(Hong Kong University of Science and Technology\, Hong Kong) as part of C
ombinatorial and additive number theory seminar (CANT 2026)\n\nLecture hel
d in Science Center in the CUNY Graduate Center (4th floor).\n\nAbstract\n
Understanding the additive structure of multiplicatively defined sets\, in
cluding the primes\, perfect squares\, perfect powers\, powerful numbers\,
and smooth numbers\, remains a fundamental open challenge. In this talk\,
I will talk about some recent progress on related results. In particular\
, I will discuss arithmetic progressions\, sumsets\, and Hilbert cubes in
some well-studied multiplicatively defined sets\, as well as their interac
tions. Joint work with Ernie Croot and Junzhe Mao.\n
LOCATION:https://researchseminars.org/talk/CANT2026/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshat Mudgal (University of Warwick\, UK)
DTSTART:20260718T150000Z
DTEND:20260718T152500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/82
DESCRIPTION:Title: A structure theorem for sets with doubling 4 + δ\nby Akshat Mudgal
(University of Warwick\, UK) as part of Combinatorial and additive number
theory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY G
raduate Center (4th floor).\n\nAbstract\nA question of Ben Green asks whet
her every finite set $A$ of integers with doubling constant $K$ must conta
in a subset $A'$ of comparable size whose doubling is at most $K + o(1)$ d
ue to some explicit algebraic structure on $A'$. This was previously under
stood in the regime $K < 4 - o(1)$ by work of Eberhard\, Green\, and Manne
rs\, who showed that one can find such a subset $A'$ with density at least
$1/2 + o(1)$ inside a long arithmetic progression. In this talk\, I will
provide a brief survey of this question as well as mention some new progre
ss towards this. This is joint work with Yifan Jing.\n
LOCATION:https://researchseminars.org/talk/CANT2026/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Cantarini (University of Perugia\, Italy)
DTSTART:20260718T153000Z
DTEND:20260718T155500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/83
DESCRIPTION:Title: Averages of the diagonal Elliott-Halberstam problem twisted by the Möbi
us function with Sobolev and Hölder-Zygmund weights\nby Marco Cantari
ni (University of Perugia\, Italy) as part of Combinatorial and additive n
umber theory seminar (CANT 2026)\n\nLecture held in Science Center in the
CUNY Graduate Center (4th floor).\n\nAbstract\nRecalling that the so-calle
d Elliott-Halberstam conjecture twisted\nby the Möbius function $\\mu(n)$
claims that $$\\sum_{q\\leq N^{\\theta}}\\max_{y\\leq N}\\max_{(a\,q)=1}\
\left|\\sum_{\\underset{{\\scriptstyle n\\equiv a\\\,\\mod\\\,q}}{n\\leq y
}}\\Lambda(n)\\mu\\left(N-n\\right)-\\frac{1}{\\varphi\\left(q\\right)}\\s
um_{n\\leq y}\\Lambda(n)\\mu\\left(N-n\\right)\\right|\\ll\\frac{N}{\\log\
\left(N\\right)^{A}} \n$$\nfor every $A>0$\, where $0<\\theta<1$ is fixed\
, and also recalling\nthat the validity of this conjecture\, in combinatio
n with the validity\nof the classical Elliott-Halberstam for suitable $\\t
heta$\, proves\nthe binary Goldbach conjecture\, in this talk we analyze w
eighted average\nvariants of this problem. We will show that\, under Gener
alized Riemann\nHypothesis\, a weak version of the Gonek-Hejhal conjecture
and working\nwith weights belonging to the Sobolev space $W^{2\,1}$ or in
the Hölder-Zygmund\nspaces $\\mathcal{C}^{\\delta}$ for suitable range o
f $\\delta$\, the\nbound of the average is consistent with the bound of th
e ``diagonal\nversions'' of this conjecture (that is\, taking $y=N$ and ta
king\n$n\\equiv N\\mod q)$. In particular\, in the case of weights in Sobo
lev\nspace\, the consistent upper bound holds for the whole $0<\\theta<1$\
nand\, in the case of weights in the Hölder-Zygmund class $\\mathcal{C}^{
\\delta}$\,\nfor $\\theta$ that depends on the choice of $\\delta$ but sti
ll not\nbelow the $1/2-2\\varepsilon$ threshold.\n
LOCATION:https://researchseminars.org/talk/CANT2026/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aviral Srivastava (IIT Gandhinagar\, India)
DTSTART:20260718T160000Z
DTEND:20260718T162500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/84
DESCRIPTION:Title: Non-Rascoe partitions and a rank parity function associated to the Roger
s–Ramanujan partitions\nby Aviral Srivastava (IIT Gandhinagar\, Indi
a) as part of Combinatorial and additive number theory seminar (CANT 2026)
\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor)
.\n\nAbstract\nThe rank-parity function associated with a class of partiti
ons gives rise to objects with rich analytical and combinatorial propertie
s. In this talk\, we will discuss the rank-parity function associated with
Rogers-Ramanujan partitions and show their close correspondence with an i
nteresting class of restricted partitions\, namely\, partitions into disti
nct parts where the number of parts is not a part. We will show some inter
esting congruences related to these functions. This talk is based on joint
\nwork with Gaurav Kumar and Prof. Atul Dixit.\n
LOCATION:https://researchseminars.org/talk/CANT2026/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiseok Yeon (University of California - Davis)
DTSTART:20260718T170000Z
DTEND:20260718T172500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/85
DESCRIPTION:Title: On the density of rational lines on cubic diagonal hypersurfaces\nby
Kiseok Yeon (University of California - Davis) as part of Combinatorial a
nd additive number theory seminar (CANT 2026)\n\nLecture held in Science C
enter in the CUNY Graduate Center (4th floor).\n\nAbstract\nIn this paper\
, we establish the asymptotic estimates for the rational lines on diagonal
cubic hypersurfaces defined by $\\sum_{i=1}^sc_ix^3_i=0$ with $c_i\\in\\m
athbb{Z}\\setminus \\{0\\}\,$ provided that $s\\geq 18.$ This improves the
previously known bound $s\\geq 21$ required to obtain such asymptotic est
imates. Our approach develops a multidimensional shifting variables argume
nt\, and exploits the recent progress on the Parsell-Vinogradov system. Th
is talk is based on the speaker’s recent work and a joint work with Pars
ell.\n
LOCATION:https://researchseminars.org/talk/CANT2026/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florian Luca (Stellenbosch University\, South Africa)
DTSTART:20260718T173000Z
DTEND:20260718T175500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/86
DESCRIPTION:Title: Multiply gleeful numbers\nby Florian Luca (Stellenbosch University\,
South Africa) as part of Combinatorial and additive number theory seminar
(CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Center
(4th floor).\n\nAbstract\nFor positive integers $k$ and $n$ let $f_k(n)$
be the number of ways of representing $n$ as a sum of $k$ powers of consec
utive primes. A number is called $k$-gleeful if \n$f_k(n)>0$ and multiply
gleeful if $f_k(n)>1$ or $f_k(n)f_{k'}(n)>0$ for some positive integers $k
< k'.$ Under Schinzel's hypothesis H\, we show that there are infinitely m
any positive integers $n$ such that $f_2(n)f_4(n)>0$. Under the same assum
ption we show that $\\limsup_{n\\to\\infty} f_2(n)=\\infty$. This gives a
conditional proof of a stronger version of a conjecture of Moore and Soren
son from the preprint arXiv:2507.09012v1\, July 2025.\n
LOCATION:https://researchseminars.org/talk/CANT2026/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francis Atta Howard (University of Abomey-Calavi\, Benin Republic)
DTSTART:20260718T180000Z
DTEND:20260718T182500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/87
DESCRIPTION:Title: Gertsch quotient living in the “poor man’s adele ring” A: Kurepa-B
ell-Wilson congruence\nby Francis Atta Howard (University of Abomey-Ca
lavi\, Benin Republic) as part of Combinatorial and additive number theory
seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduat
e Center (4th floor).\n\nAbstract\nWilson's theorem is related to left fac
torials\, expressed as $K_p \\equiv \\mathbf{Bell}_{p-1} - 1 \\pmod p$\, f
or prime $p\\geq3$. This study examines a Kurepa-Bell-Wilson congruence (K
BW)\, $$\\frac{K_p + 1}{p}\\equiv \\frac{ \\mathbf{Bell}_{p-1}}{p}+ W_p \\
pmod{p}\,$$ and demonstrates that it naturally generates the non-zero "Ger
tsch quotient ($\\mathbb{G}_p$)\," which\, for larger primes modulo $p$ re
sides in the poor man's adele ring $\\mathcal{A}$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bobby Jacobs
DTSTART:20260718T183000Z
DTEND:20260718T185500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/88
DESCRIPTION:Title: Primes in the Fibonacci ring\nby Bobby Jacobs as part of Combinatori
al and additive number theory seminar (CANT 2026)\n\nLecture held in Scien
ce Center in the CUNY Graduate Center (4th floor).\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/CANT2026/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Augustine O. Munagi (University of the Witwatersrand\, South Afric
a)
DTSTART:20260718T190000Z
DTEND:20260718T192500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/89
DESCRIPTION:Title: A Bessenrodt-Ono inspired inequality for compositions proved constructiv
ely\nby Augustine O. Munagi (University of the Witwatersrand\, South A
frica) as part of Combinatorial and additive number theory seminar (CANT 2
026)\n\nLecture held in Science Center in the CUNY Graduate Center (4th fl
oor).\n\nAbstract\nIn 2016 Bessenrodt-Ono published an analytic proof of t
he inequality $p(a+b)\\leq p(a)p(b)$\, where $p(n)$ is the partition funct
ion and $a\,b$ are positive integers with $a+b>8$. In this talk we conside
r a similar result for $c(n)$\, the number of integer compositions of $n$\
, and show that $c(a+b)>c(a)c(b)$ for all positive integers $a\,b$. Beside
s numerical verifications\, we provide a constructive bijective proof base
d on the inherent symmetry of compositions. It is known that such a proof
is still elusive in the partitions case. We also give an application of ou
r machinery to efficient generation of compositions.\n
LOCATION:https://researchseminars.org/talk/CANT2026/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurence P. Wijaya (University of Kentucky)
DTSTART:20260718T193000Z
DTEND:20260718T195500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/90
DESCRIPTION:Title: Polynomial corners over finite field\nby Laurence P. Wijaya (Univers
ity of Kentucky) as part of Combinatorial and additive number theory semin
ar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate Cent
er (4th floor).\n\nAbstract\nWe study the size of subset $\\mathbf{F}_p$ w
ith $p$ primes goes to infinity not containing a configuration of the form
$(x\,y)\,(x+P(z)\,y)\,(x\,y+P(z))$ for some polynomial $P$ of degree at l
east $3$. Similar result was found by Kravitz\, Kuca\, and Leng for the in
teger setting under some conditions. Our result provides better bound in t
he $\\mathbf{F}_p$ setting.\n
LOCATION:https://researchseminars.org/talk/CANT2026/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Glenn T. Bruda (University of Florida)
DTSTART:20260718T200000Z
DTEND:20260718T202500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/91
DESCRIPTION:Title: Generalized polygonal number representations\nby Glenn T. Bruda (Uni
versity of Florida) as part of Combinatorial and additive number theory se
minar (CANT 2026)\n\nLecture held in Science Center in the CUNY Graduate C
enter (4th floor).\n\nAbstract\nFor $k\\geq5$ and $n\\geq 4$\, let $r_n^{(
k)}(N)$ be the number of representations of $N$ as the sum of $n$ generali
zed $k$-gonal numbers and $r_n^{\\square}(N)$ be the number of representat
ions of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle
method\, we prove a closed-form asymptotic relation between $r_{n}^{(k)}(
N)$ and $r_n^{\\square}(N)$ for $k\\not\\equiv 0\\bmod 4$ and any $n\\geq4
$. Consequently\, we relate the number of representations of $N$ as the su
m of four ordinary $k$-gonal numbers to $r_4^{\\square}(N)$ via a result o
f Bringmann--Jang--Kane--Tse.\n
LOCATION:https://researchseminars.org/talk/CANT2026/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Collier Gaiser (Community College of Aurora\, Colorado)
DTSTART:20260718T203000Z
DTEND:20260718T205500Z
DTSTAMP:20260710T091017Z
UID:CANT2026/92
DESCRIPTION:Title: On Rado numbers for equations with unit fractions\nby Collier Gaiser
(Community College of Aurora\, Colorado) as part of Combinatorial and add
itive number theory seminar (CANT 2026)\n\nLecture held in Science Center
in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $R_r(k)$ be the
smallest $n$ such that every $r$-coloring of $\\{1\,2\,...\,n\\}$ has a mo
nochromatic solution to $x_1+x_2+\\cdots+x_k=y$\, where $x_1\,x_2\,\\ldots
\,x_k$ are not necessarily distinct. Beutelspacher and Brestovansky proved
that $R_2(k)=k^2+k-1$ and\, recently\, Boza\, Mar\\'{i}n\, Revuelta\, and
Sanz proved that $R_3(k)=k^3+2k^2-2$. Similarly\, let $f_r(k)$ be the sma
llest $n$ such that every $r$-coloring of $\\{1\,2\,...\,n\\}$ has a monoc
hromatic solution to the equation $1/x_1+1/x_2+\\cdots+1/x_k=1/y$\, where
$x_1\,x_2\,\\ldots\,x_k$ are not necessarily distinct. Brown and R\\"{o}dl
proved that $f_2(k)=O(k^6)$. In this talk\, we show that $f_2(k)=O(k^3)$
and $f_3(k)=O(k^{43})$. The main ingredient in our proof is a finite set $
A\\subseteq\\mathbb{N}$ such that every $r$-coloring of $A$ has a monochro
matic solution to the linear equation $x_1+x_2+\\cdots+x_k=y$ and the leas
t common multiple of $A$ is sufficiently small. As for the lower bound\, w
e show that $f_r(k)\\geq k^r$ which leads to an interesting open question:
Is $f_2(k)=\\Theta(k^2)$?\n
LOCATION:https://researchseminars.org/talk/CANT2026/92/
END:VEVENT
END:VCALENDAR