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BEGIN:VEVENT
SUMMARY:Piotr Miska (Jagiellonian University in Krak\\'{o}w\, Poland)
DTSTART:20240524T130000Z
DTEND:20240524T132500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/1
DESCRIPTION:Title: On the Frobenius problem with restrictions on common
divisors of coefficients\nby Piotr Miska (Jagiellonian University in K
rak\\'{o}w\, Poland) as part of Combinatorial and additive number theory (
CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate
Center.\n\nAbstract\nLet $m\,s\,t$ be positive integers with $t\\leq s-2$
and let $a_1\,a_2\,\\ldots\,a_s$ be positive integers such that $(a_1\,a_
2\,\\ldots\,a_{s-1})=1$. In the paper we prove that every sufficiently lar
ge positive integer can be written in the form $a_1\\mu_1+a_2\\mu_2+\\ldot
s+a_s\\mu_m$\, where the positive integers $\\mu_1\,\\mu_2\,\\ldots\,\\mu
_s$ have no common divisor that is the $m$-th power of a positive integer
greater than $1$\, but each $t$ of the values $\\mu_1\,\\mu_2\,\\ldots\,\
\mu_s$ do have a common divisor that is the $m$-th power of a positive int
eger greater than $1$. Moreover\, we show that every sufficiently large po
sitive integer can be written as a sum of positive integers $\\mu_1\,\\mu_
2\,\\ldots\,\\mu_s$ with no common divisor that is the $m$-th power of a p
ositive integer greater than $1$\, but each $s-1$ of the values of $\\mu_1
\,\\mu_2\,\\ldots\,\\mu_s$ do have a common divisor that is the $m$-th pow
er of a positive integer greater than $1$. \n\nJoint work with Maciej Zaka
rczemny (Cracow University of Technology).\\\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:I.D. Shkredov (Purdue University)
DTSTART:20240524T133000Z
DTEND:20240524T135500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/2
DESCRIPTION:Title: On universal sets and sumsets\nby I.D. Shkredov (
Purdue University) as part of Combinatorial and additive number theory (CA
NT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate C
enter.\n\nAbstract\nLet $G$ be an abelian group. A set $A \\subseteq G$ is
called \na {\\it $k$--universal} set if for any $x_1\,\\dots\,x_k \\in G
$ there exists $s\\in G$ \nsuch that $x_1+s\,\\dots\,x_k+s \\in A$. The te
rm ``universal set'' was introduced \nby Alon\, Bukh\, and Sudakov in conn
ection with the discrete Kakeya problem. \nWe study the concept of univers
al sets from the additive--combinatorial point of view. Among other resul
ts we obtain some applications of this type of uniformity to sets avoiding
solutions to linear equations\, and get an optimal upper bound for the co
vering number of general sumsets.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Beker (University of Zagreb\, Croatia)
DTSTART:20240524T140000Z
DTEND:20240524T142500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/3
DESCRIPTION:Title: On a problem of Erdos and Graham about consecutive su
ms in strictly increasing sequences\nby Adrian Beker (University of Za
greb\, Croatia) as part of Combinatorial and additive number theory (CANT
2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Cent
er.\n\nAbstract\nGiven a finite sequence of integers $a = (a_i)_{1\\leq i
\\leq k}$\, let $S(a)$ denote the set of its consecutive sums\, that is\,
sums of the form $\\sum_{i=u}^{v}a_i$ with $1 \\leq u \\leq v \\leq k$. E
rd\\H os and Graham asked whether there exists a constant $c > 0$ such tha
t\, for all positive integers $n$\, there is such a sequence in $\\{1\,\\l
dots\,n\\}$ which is strictly increasing and satisfies $|S(a)| \\geq cn^2$
. \n\nThe obvious candidate consisting of all integers from $1$ up to $n$
falls short of having this property due to reasons related to the multipli
cation table problem. On the other hand\, if we drop the monotonicity assu
mption\, such sequences were shown to exist by Hegyv\\'ari via a construc
tion based on Sidon sets. In this talk\, I will present two constructions
\, one probabilistic and the other deterministic\, that give an affirmativ
e answer to the starting question. I will also discuss some non-trivial u
pper bounds on the size of $S(a)$ in this setting.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Chapman (University of Bristol\, UK)
DTSTART:20240524T143000Z
DTEND:20240524T145500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/4
DESCRIPTION:Title: Monochromatic sums and products\nby Jonathan Chap
man (University of Bristol\, UK) as part of Combinatorial and additive num
ber theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the C
UNY Graduate Center.\n\nAbstract\nIf we colour $\\{2\,\\ldots\,N\\}$ with
$r$ different colours\, how many monochromatic solutions to $xy=z$ appear?
A classical theorem of Schur shows that we always obtain $(c_r+o(1))N^2$
monochromatic solutions (as $N\\to\\infty$) to $x+y=z$\, for some $c_r>0$\
, which is within a constant factor of the total number of solutions. Howe
ver\, Prendiville showed that one cannot achieve such a strong result for
$xy=z$\, even if one only uses $2$ colours.\n \n In this talk\, I will pre
sent recent work on determining the asymptotic minimum number of monochrom
atic solutions to $xy=z$. We prove that every $2$-colouring of $\\{2\,\\ld
ots\,N\\}$ produces \n at least $(2^{-3/2} + o(1))\\sqrt{N}\\log N$ monoch
romatic solutions to $xy=z$\, and the leading constant is sharp. I will al
so introduce a Schur-type problem for colourings of real numbers. If the d
iscrete and continuous Schur problems are `quantitatively equivalent'\, th
en\, for an arbitrary number of colours\, our upper and lower bounds for t
he number of monochromatic solutions to $xy=z$ match up to a logarithmic f
actor.\n \n Joint work with Lucas Aragao\, Miquel Ortega\, and Victor Souz
a.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrzej Kukla (Jagiellonian University in Krak\\'{o}w\, Poland)
DTSTART:20240524T150000Z
DTEND:20240524T152500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/5
DESCRIPTION:Title: Representing the number of binary partitions as sums
of three squares\nby Andrzej Kukla (Jagiellonian University in Krak\\
'{o}w\, Poland) as part of Combinatorial and additive number theory (CANT
2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Cent
er.\n\nAbstract\nLet $c_m(n)$ denote the number of partitions of $n$ into
parts that are powers of 2 such that part equal to 1 takes one among $2m$
colors and each part $>1$ takes one among $m$ colors. The study of this fu
nction was initiated in 2021 by Zmija and Ulas\, who focused on its 2-adic
behaviour. In particular\, they found a formula for the 2-adic valuation
of $c_m(n)$ that depends on the 2-adic valuation of $m$ and the value of $
t_n - t_{n-1}$\, where $t_n$ is the $n$-th term of the Prouhet-Thue-Morse
sequence. During the talk we will be considering the diophantine equation
$c_m(n) = x^2 + y^2 + z^2$. For fixed $m$\, we will characterize the set o
f natural numbers $n$\, for which the solution does not exist\, and then f
urther investigate properties of these sets. The talk is based on an ongoi
ng work on speaker's master's thesis.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Technau (Paderborn University\, Germany)
DTSTART:20240524T153000Z
DTEND:20240524T155500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/6
DESCRIPTION:Title: Cilleruelo's conjecture on the LCM of polynomial sequ
ences\nby Marc Technau (Paderborn University\, Germany) as part of Com
binatorial and additive number theory (CANT 2024)\n\nLecture held in Room
4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nWe discuss a
conjecture of Cilleruelo on the growth of the least common multiple of con
secutive values of a polynomial and subsequent progress towards it in work
of Maynard--Rudnick and Sah.\nIn recent work\, the speaker and\, independ
ently\, Alexei Entin made further advances by exploiting symmetries amongs
t the roots of the polynomials in question.\nWe shall discuss these approa
ches and related beautiful work of Baier and Dey.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bartosz Sobolewski (Jagiellonian University\, Krak\\'{o}w\, Poland
and Montanuniversit{\\"a}t Leoben\, Austria)
DTSTART:20240524T173000Z
DTEND:20240524T175500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/8
DESCRIPTION:Title: On block occurrences in the binary expansions of $n$
and $n+t$\nby Bartosz Sobolewski (Jagiellonian University\, Krak\\'{o}
w\, Poland and Montanuniversit{\\"a}t Leoben\, Austria) as part of Combina
torial and additive number theory (CANT 2024)\n\nLecture held in Room 4102
and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $s(n)$ denote
the sum of binary digits of a nonnegative integer $n$. In the recent year
s there has been significant progress concerning the behavior of the diffe
rences $s(n+t)-s(n)$\, where $t$ is a fixed nonnegative integer. In partic
ular\, Spiegelhofer and Wallner proved that for $t$ having sufficiently ma
ny blocks $01$ in its binary expansion\, the set $\\{n: s(n+t) \\geq s(n)\
\}$ has natural density $> 1/2$ (partially confirming a conjecture by Cusi
ck). Moreover\, for such $t$ the distribution $s(n+t) - s(n)$ is close to
Gaussian. During the talk we consider an analogue of this problem concerni
ng the function $r(n)$\, which counts the occurrences of the block $11$ in
the binary expansion of $n$. In particular\, we prove that the distribut
ion of $r(n+t)-r(n)$ is approximately Gaussian as well. We also discuss a
generalization to an arbitrary block of binary digits.\n\nJoint work with
Lukas Spiegelhofer (Montanuniversit{\\"a}t Leoben).\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Alfr\\' ed R\\' enyi Institute of Mathematics\, Buda
pest\, Hungary)
DTSTART:20240524T180000Z
DTEND:20240524T182500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/9
DESCRIPTION:Title: Solutions to the discrete Pompeiu problem and to the
finite Steinhaus tiling problem\nby Gergely Kiss (Alfr\\' ed R\\' enyi
Institute of Mathematics\, Budapest\, Hungary) as part of Combinatorial a
nd additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Roo
m 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $K$ be a nonempty fin
ite subset of the Euclidean space $\\mathbb{R}^k$ $(k\\ge 2)$.\nIn this ta
lk we discuss the solution of the following so-called discrete Pompeiu pro
blem. If a function $f\\colon \\mathbb{R}^k \\to \\mathbb{C}$ is such that
the sum of $f$\non every congruent copy of $K$ is zero\, then $f$ vanishe
s everywhere. In fact\, we solve\na stronger\, weighted version of this pr
oblem. As a corollary we obtain that every\nfinite subset of $\\mathbb{R}^
k$ having at least two elements is a Jackson\nset\; that is\, no subset of
$\\mathbb{R}^k$ intersects every congruent copy of $K$ in\nexactly one po
int.\n\nJoint work with Mikl\\' os Laczkovich.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Cumberbatch (Purdue University)
DTSTART:20240524T183000Z
DTEND:20240524T185500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/10
DESCRIPTION:Title: Digitally restricted sets and the Goldbach conjectur
e\nby James Cumberbatch (Purdue University) as part of Combinatorial a
nd additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Roo
m 9207 in the CUNY Graduate Center.\n\nAbstract\nWe show that given any ba
se $b$ and any set of digits $\\mathcal{D}$ with at least two digits\, let
$\\mathcal{A}$ be the set of integers whose base-$b$ digits consist only
of values in $\\mathcal{D}$. We prove that almost all even integers in $\\
mathcal{A}$ are sum of two primes.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/10
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mel Nathanson (Lehman College (CUNY))
DTSTART:20240524T190000Z
DTEND:20240524T192500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/11
DESCRIPTION:Title: Polynomial equations in infinitely many variables\nby Mel Nathanson (Lehman College (CUNY)) as part of Combinatorial and a
dditive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 92
07 in the CUNY Graduate Center.\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/11
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mizan Khan (Eastern Connecticut State University)
DTSTART:20240524T193000Z
DTEND:20240524T195500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/12
DESCRIPTION:Title: A conjecture for clean lattice parallelograms\nb
y Mizan Khan (Eastern Connecticut State University) as part of Combinatori
al and additive number theory (CANT 2024)\n\nLecture held in Room 4102 and
Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $P \\subseteq {\\
mathbb R}^2$ be a convex lattice polygon containing at least one lattice p
oint in its interior. The interior hull of $P$\, denoted by $P^{(1)}$\, is
the convex closure of the set of lattice points in the interior of $P$\,
that is\,\n$$P^{(1)} = \\operatorname{conv}\\left(\\operatorname{interior}
(P) \\cap {\\mathbb Z}^2\\right).$$\nWe can now form a finite nested seque
nce of interior hulls\n$$P^{(1)}\\supseteq P^{(2)} \\supseteq P^{(3)} \\su
pseteq \\ldots\, $$\nwhere $P^{(2)}$ is the interior hull of $P^{(1)}$\,
$P^{(3)}$ is the interior hull of $P^{(2)}$\, and so on. \n\nWe will prese
nt some experimental data supporting a conjecture on the average number of
interior hulls for clean lattice parallelograms. (A lattice parallelogra
m is said to be clean if the only lattice points on its boundary are the 4
vertices.) \n\nJoint work with Riaz Khan.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/12
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathaniel Kingsbury (CUNY Graduate Center)
DTSTART:20240524T200000Z
DTEND:20240524T202500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/13
DESCRIPTION:Title: The square-root law does not hold in the presence of
zero divisors\nby Nathaniel Kingsbury (CUNY Graduate Center) as part
of Combinatorial and additive number theory (CANT 2024)\n\nLecture held in
Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $R$
be a finite ring and define the paraboloid $P = \\{(x_1\, \\dots\, x_d)\\
in R^d|x_d = x_1^2 + \\dots + x_{d-1}^2\\}.$ Suppose that for a sequence o
f finite rings of size tending to infinity\, the Fourier transform of $P$
satisfies a square-root type bound constant $C$. Then all but finitely man
y of the rings are fields.\n\nMost of our argument works in greater genera
lity: let $f$ be a polynomial with integer coefficients in $d-1$ variables
\, with a fixed order of variable multiplications (so that it defines a fu
nction $R^{d-1}\\rightarrow R$ even when $R$ is noncommutative)\, and set
$V_f = \\{(x_1\, \\dots\, x_d)\\in R^d|x_d = f(x_1\, \\dots\, x_{d-1})\\}
$. If (for a sequence of finite rings of size tending to infinity) we have
a square-root type bound on the Fourier transform of $V_f$\, then all but
finitely many of the rings are fields or matrix rings of small dimension.
\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/13
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20240524T203000Z
DTEND:20240524T205500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/14
DESCRIPTION:Title: Multi-parameter point-line incidence estimates in fi
nite fields and applications\nby Steven Senger (Missouri State Univers
ity) as part of Combinatorial and additive number theory (CANT 2024)\n\nLe
cture held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbst
ract\nWe present some novel multi-parameter point-line incidence estimates
in vector spaces over finite fields. These outperform the straightforward
higher-dimensional analogs. We focus on an application to sums and produc
ts type problems.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/14
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fred Tyrrell (University of Bristol\, UK)
DTSTART:20240522T153000Z
DTEND:20240522T155500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/15
DESCRIPTION:Title: New lower bounds for cap sets\nby Fred Tyrrell (
University of Bristol\, UK) as part of Combinatorial and additive number t
heory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY G
raduate Center.\n\nAbstract\nA cap set is a subset of $\\mathbb{F}_3^n$ wi
th no solutions to $x+y+z=0$ other than when $x=y=z$. The cap set problem
asks how large a cap set can be\, and is an important problem in additive
combinatorics and combinatorial number theory. In this talk\, I will intro
duce the problem\, give some background and motivation\, and describe how
I was able to provide the first progress in 20 years on the lower bound fo
r the size of a maximal cap set. Building on a construction of Edel\, we u
se improved computational methods and new theoretical ideas to show that\,
for large enough $n$\, there is always a cap set in $\\mathbb{F}_3^n$ of
size at least $2.218^n$. I will then also discuss recent developments\, in
cluding an extension of this result by Google DeepMind.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/15
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:G\\'abor Somlai (E\\"otv\\"os Lor\\' and University and R\\'enyi I
nstitute\, Hungary)
DTSTART:20240522T130000Z
DTEND:20240522T132500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/16
DESCRIPTION:Title: New method for old results of R\\'edei\, Lov\\'asz a
nd Schrijver\nby G\\'abor Somlai (E\\"otv\\"os Lor\\' and University a
nd R\\'enyi Institute\, Hungary) as part of Combinatorial and additive num
ber theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in the C
UNY Graduate Center.\n\nAbstract\nR\\'edei proved that a set $S$ of cardin
ality $p$ in $\\mathbb{F}_p^2$ determines at least $\\frac{p+3}{2}$ direct
ions or $S$ is a line. \nWe managed find a short proof for R\\'edei's res
ult avoiding the theory of lacunary polynomials by proving the following s
tatement. Let $f $ be a polynomial over the finite field $\\mathbb{F}_p$.
Consider the elements of the range as integers in $\\{0\,1\, \\ldots\, p-1
\\}$. Assume that $\\sum_{x \\in \\mathbb{F}_p}f(x)=p$. Then either $f=1$
or $deg(f) \\ge \\frac{p-1}{2}$.\nThe uniqueness (up to affine transforma
tions) of the sets of size $p$\nin $\\mathbb{F}_p^2$ was proved by Lov\\'a
sz and Schrijver. The same result follows from the almost uniqueness of th
e polynomials of degree $\\frac{p-1}{2}$ of range sum $p$.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/16
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Konyagin (Steklov Institute of Mathematics\, Russia)
DTSTART:20240522T133000Z
DTEND:20240522T135500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/17
DESCRIPTION:Title: On distinct angles in the plane\nby Sergei Konya
gin (Steklov Institute of Mathematics\, Russia) as part of Combinatorial a
nd additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Roo
m 9207 in the CUNY Graduate Center.\n\nAbstract\nThe talk is based on our
joint paper with Jonathan Passant and Misha Rudnev. \nWe prove that if $N$
points lie in convex position in the plane\, then they determine\n$\\gg N
^{1+3/23+o(1)}$ distinct angles\, provided no $N-1$ points lie on a common
circle.\nThis is the first super--linear bound on the distinct angle prob
lem that has received\nrecent attention.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/17
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krystian Gajdzica (Jagiellonian University\, Krak\\' ow\, Poland)
DTSTART:20240522T140000Z
DTEND:20240522T142500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/18
DESCRIPTION:Title: A combinatorial approach to the Bessenrodt-Ono type
inequalities\nby Krystian Gajdzica (Jagiellonian University\, Krak\\'
ow\, Poland) as part of Combinatorial and additive number theory (CANT 202
4)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Center.
\n\nAbstract\nIn 2016\, Bessenrodt and Ono showed that the partition funct
ion satisfies the inequality of the form\n $$p(a)p(b)>p(a+b)$$\nfor all
$a\,b\\geqslant2$ with $a+b>9$. Their proof is based on the asymptotic es
timates of $p(n)$ due to Lehmer. Since then\, a lot of similar phenomena h
ave been discovered for various variations of the partition function. \n\n
We discuss the analogue of the Bessenrodt-Ono inequality for the so-called
$A$-partition function $p_A(n)$\, which enumerates those partitions of $n
$ whose parts belong to a fixed set $A\\subset\\mathbb{N}$. Since there is
no known asymptotic formula for $p_A(n)$ in general\, we can not deal wit
h the problem using any estimates of $p_A(n)$. Therefore\, we present a co
mbinatorial approach to the issue by constructing an appropriate injection
between some sets of partitions.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/18
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rauan Kaldybayev (Williams College)
DTSTART:20240522T150000Z
DTEND:20240522T152500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/19
DESCRIPTION:Title: Limiting behavior in missing sums of sumsets\nby
Rauan Kaldybayev (Williams College) as part of Combinatorial and additive
number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in t
he CUNY Graduate Center.\n\nAbstract\nWe study $|A + A|$ as a random varia
ble\, where $A \\subseteq \\{0\, \\dots\, N\\}$ is a random subset such\n
that each\n $0 \\le n \\le N$ is included with probability $0 < p < 1$\,
and where $A + A$ is the set of sums $a + b$ for $a\,b$\n in $A$.\n Lazar
ev\, Miller\, and O'Bryant studied the distribution of $2N + 1 - |A + A|$\
, the number of summands not\n represented in\n $A + A$ when $p = 1/2$. A
recent paper by Chu\, King\, Luntzlara\, Martinez\, Miller\, Shao\, Sun\,
and Xu generalizes\n this to\n all $p\\in (0\,1)$\, calculating the firs
t and second moments of the number of missing summands and establishing\n
exponential upper and lower bounds on the probability of missing exactly
$n$ summands\, mostly working in the\n limit of\n large $N$. We provide e
xponential bounds on the probability of missing at least $n$ summands\, fi
nd another\n expression\n for the second moment of the number of missing
summands\, extract its leading-order behavior in the limit of\n small $p$\
,\n and show that the variance grows asymptotically slower than the mean\
, proving that for small $p$\, the number of\n missing\n summands is very
likely to be near its expected value.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/19
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian T\\'afula (Universit\\'e de Montr\\'eal\, Canada)
DTSTART:20240522T160000Z
DTEND:20240522T162500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/20
DESCRIPTION:Title: Representation functions with prescribed rates of gr
owth\nby Christian T\\'afula (Universit\\'e de Montr\\'eal\, Canada) a
s part of Combinatorial and additive number theory (CANT 2024)\n\nLecture
held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\n
Let $h\\geq 2$\, and $b_1$\, $\\ldots$\, $b_h$ be positive integers with $
\\gcd = 1$. For a set $A\\subseteq \\mathbb{N}$\, denote by $r_A(n)$ the n
umber of solutions to the equation\n \\[ b_1 k_1 + ... + b_h k_h = n \\]\n
with $k_1$\, $\\ldots$\, $k_h\\in A$. For which functions $F$ can we find
$A$ such that $r_A(n) \\sim F(n)$? Or $r_A(n)\\asymp F(n)$? In the asympt
otic case\, we show that for every $F$ of regular variation satisfying\n \
\[ \\frac{F(x)}{\\log x} \\xrightarrow{x\\to\\infty} \\infty\, \\quad\\tex
t{and}\\quad F(x) \\leq (1 + o(1)) \\dfrac{x^{h-1}}{(h-1)!b_1...b_h}\, \\]
\n there is $A$ such that $r_A(n) \\sim F(n)$. In the order of magnitude c
ase\, there is $A$ with $r_A(n)\\asymp F(x)$ for every $F$ non-decreasing
such that $F(2x)\\ll F(x)$ in the range $\\log x \\ll F(x) \\ll x^{h-1}$.
This extends earlier work of Erdős–Tetali and Vu\, and addresses a ques
tion raised by Nathanson on which functions can be the $r_A$ of some $A$.\
n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/20
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Lebowitz-Lockard (University of Texas\, Tyler\, TX)
DTSTART:20240522T173000Z
DTEND:20240522T175500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/21
DESCRIPTION:Title: On the smallest parts of partitions into distinct pa
rts\nby Noah Lebowitz-Lockard (University of Texas\, Tyler\, TX) as pa
rt of Combinatorial and additive number theory (CANT 2024)\n\nLecture held
in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nFor
a given integer $n$\, let $D(n)$ be the set of partitions of $n$ into dist
inct parts. Create a sum as follows. For each partition $\\lambda$ in $D(n
)$\, add the smallest element of $\\lambda$ if it is even and subtract it
if it is odd. A classic theorem of Uchimura states that this quantity is e
qual to the number of divisors of $n$. We generalize this result to the su
m of the $k$th smallest elements of partitions for a fixed value of $k$. W
e also consider some further generalizations\, as well as variants for the
smallest number not in a given partition. \\\\\nJoint work with Rajat Gup
ta and Joseph Vandehey.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/21
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Donley (Queensborough Community College (CUNY))
DTSTART:20240522T180000Z
DTEND:20240522T182500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/22
DESCRIPTION:Title: A combinatorial introduction to adinkras\nby Rob
ert Donley (Queensborough Community College (CUNY)) as part of Combinatori
al and additive number theory (CANT 2024)\n\nLecture held in Room 4102 and
Room 9207 in the CUNY Graduate Center.\n\nAbstract\nIn 2005\, Faux and Ga
tes defined the adinkra\, a graphical device for describing particle excha
nges in supersymmetry. Independent of the physical applications\, the adi
nkra resides at a nexus of various mathematical concepts and problems. In
this talk\, we give an introduction to adinkras from the point of view of
matching problems in combinatorics. \n\nJoint work with S. James Gates\,
Jr.\, Tristan H{\\"u}bsch\, and Rishi Nath.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/22
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Flores (Purdue University)
DTSTART:20240522T183000Z
DTEND:20240522T185500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/23
DESCRIPTION:Title: A circle method approach to K-multimagic squares
\nby Daniel Flores (Purdue University) as part of Combinatorial and additi
ve number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in
the CUNY Graduate Center.\n\nAbstract\nWe investigate $K$-multimagic squa
res of order $N$\, which are $N \\times N$ magic squares with remain magic
after raising each element to the $k$th power for all $2 \\le k \\le K$.
Given $K \\ge 2$\, we consider the problem of establishing the smallest in
teger $N(K)$ for which there exists nontrivial $K$-multimagic squares of o
rder $N(K)$. Previous results on multimagic squares show that $N(K) \\le (
4K-2)^K$ for large $K$. Here we utilize the Hardy-Littlewood circle method
and establish the bound \n\\[N(K) \\le 2K(K+1)+1.\\] \n\nVia an argument
of Granville's we additionally deduce the existence of infinitely many \\e
mph{non-trivial} prime valued $K$-multimagic squares of order $2K(K+1)+1$.
\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/23
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonid Fel (Technion -- Israel Institute of Technology\, Israel)
DTSTART:20240522T190000Z
DTEND:20240522T192500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/24
DESCRIPTION:Title: Ratio between a sum of generators and rational power
s of their product\nby Leonid Fel (Technion -- Israel Institute of Tec
hnology\, Israel) as part of Combinatorial and additive number theory (CAN
T 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Ce
nter.\n\nAbstract\nWe study a ratio $R_m(k)\\!= I_1/\\sqrt[k]{I_m}$ betwee
n a sum $I_1\\!=\\!\n\\sum_{j=1}^md_j$ of generators \nand rational powers
$\\sqrt[k]{I_m}$ of their product $I_m=\\prod_{j=1}^md_j$ in numerical \n
semigroups $\\langle d_1\,\\ldots\,d_m\\rangle$. We find its upper ${\\sf
R}_m^+(k)$ and lower ${\\sf R}_m^-(k)$ bounds \n in the range $1\\le k\\le
m$. We prove that $R_m(k)$ has a universal upper bound \nif and only if
$m\\ge 2k-1$.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/24
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksei Volostnov (Moscow Institute of Physics and Technology\, Ru
ssia)
DTSTART:20240522T193000Z
DTEND:20240522T195500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/25
DESCRIPTION:Title: On the additive energy of roots\nby Aleksei Volo
stnov (Moscow Institute of Physics and Technology\, Russia) as part of Com
binatorial and additive number theory (CANT 2024)\n\nLecture held in Room
4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $p$ be a
prime number\, $f\\in \\mathbb F_p[x]$ be a polynomial of small degree and
a set $A\\subset \\mathbb F_p$ have sufficiently small cardinality in ter
ms of $p$. \nWe study the number of solutions to the equation (in $\\mathb
b F_p$)\n\\[\nx_1+x_2 = x_3+x_4\,\\quad f(x_1)\,f(x_2)\,f(x_3)\,f(x_4)\\in
A\,\n\\]\nprovided that $A$ has small doubling. Namely\, we improve the u
pper bound \nfrom recent work by B. Kerr\, I. D. Shkredov\, I. E. Shparlin
ski and A. Zaharescu.\n\nMoreover\, we address questions of cardinalities
$|A+A|$ vs $|f(A)+f(A)|$. \nIn particular\, we prove that \n\\[\n \\ma
x(|A+A|\,|A^3+A^3|)\\gg |A|^{16/15} \n \\] \n \\[\n \\max(|A+A|\
,|A^4 +A^4|)\\gg |A|^{25/24} \n\\]\n \\[\n \\max(|A+A|\,|A^5+A^5|)\
\gg |A|^{25/24}. \n\\]\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/25
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carl Pomerance (Dartmouth College)
DTSTART:20240522T200000Z
DTEND:20240522T202500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/26
DESCRIPTION:Title: Matchable numbers\nby Carl Pomerance (Dartmouth
College) as part of Combinatorial and additive number theory (CANT 2024)\n
\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\n
Abstract\nFor a natural number $n$ let $D(n)$ denote the set of positive d
ivisors of $n$ and\nlet $\\tau(n)=\\#D(n)$. Say $n$ is {\\it matchable} i
f there is a bijection from\n$D(n)$ to $\\{1\,2\,\\dots\,\\tau(n)\\}$ with
corresponding numbers relatively prime.\nFor example\, each number up to
7 is matchable\, but 8 is not.\nThis definition was made by Santos on Math
Overflow in 2022\; he\nasks if there are more matchable numbers \nthan not
. We prove this by showing the set of matchable numbers has an asymptotic
density\ngiven by $\\prod_{p\\\,{\\rm prime}}(1-1/p^p)=.72199\\dots$. \\
\\\nThis is joint work with Nathan McNew.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/26
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Miller (Williams College)
DTSTART:20240522T203000Z
DTEND:20240522T205500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/27
DESCRIPTION:Title: The theory of normalization constants and Zeckendorf
decompositions\nby Steve Miller (Williams College) as part of Combina
torial and additive number theory (CANT 2024)\n\nLecture held in Room 4102
and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nIf we define the
Fibonacci numbers to start 1\, 2\, 3\, 5 and so on\, we have a wonderful p
roperty: Every positive integer has a unique representation as a sum of no
n-adjacent terms. Called the Zeckendorf decomposition\, we can prove many
results about the summands\, from the number in a typical decomposition co
nverging to a Gaussian to the probabilities of gaps converging to a geomet
ric decay. Many of these proofs are straightforward but tedious exercises
in algebra. We present a new approach\, which so far has just been applie
d to the distribution of gaps\, but hopefully can work for related problem
s\, which bypasses these calculations through the theory of normalization
constants.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/27
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Sellers (University of Minnesota Duluth)
DTSTART:20240522T143000Z
DTEND:20240522T145500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/28
DESCRIPTION:Title: Elementary proofs of congruences for POND and PEND p
artitions\nby James Sellers (University of Minnesota Duluth) as part o
f Combinatorial and additive number theory (CANT 2024)\n\nLecture held in
Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nRecently
\, Ballantine and Welch considered two classes of integer partitions which
they labeled POND and PEND partitions. These are integer partitions where
in the odd parts (respectively\, the even parts) cannot be distinct. In re
cent work\, I studied these two types of partitions from an arithmetic per
spective and proved infinite families of mod 3 congruences satisfied by th
e two corresponding enumerating functions. I will talk about the generatin
g functions for these enumerating functions\, and I will also highlight th
e (induction) proofs that I utilized.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/28
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jin-Hui Fang (Nanjing Normal University\, China)
DTSTART:20240523T130000Z
DTEND:20240523T132500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/29
DESCRIPTION:Title: On Cilleruelo-Nathanson's method in Sidon sets\n
by Jin-Hui Fang (Nanjing Normal University\, China) as part of Combinatori
al and additive number theory (CANT 2024)\n\nLecture held in Room 4102 and
Room 9207 in the CUNY Graduate Center.\n\nAbstract\nFor nonnegative integ
ers $h\,g$ with $h\\ge 2$\, a set $\\mathcal{A}$ of nonnegative integers i
s defined as a $B_h[g]$ sequence if\, for every nonnegative integer $n$\,
the number of representations of $n$ with the form $n=a_1+a_2+\\cdots+a_h$
is no larger than $g$\, where $a_1\\le \\cdots \\le a_h$ and $a_i\\in \\m
athcal{A}$ for $i=1\,2\,\\cdots\,h$. Let $\\mathbb{Z}$ be the set of integ
ers and $\\mathbb{N}$ be the set of positive integers. In 2013\, by intro
ducing the method of \\emph{Inserting Zeros Transformation}\, Cilleruelo a
nd Nathanson obtained the following nice result: let $f:\\mathbb{Z}\\right
arrow \\mathbb{N}\\bigcup \\{0\,\\infty\\}$ be any function such that $\\l
iminf_{|n|\\rightarrow \\infty} f(n)\\ge g$ and let $\\mathcal{B}$ be any
$B_h[g]$ sequence. Then\, for any decreasing function $\\epsilon(x)\\right
arrow 0$ as $x\\rightarrow \\infty$\, there exists a sequence $\\mathcal{A
}$ of integers such that $r_{\\mathcal{A}\,h}(n)=f(n)$ for all $n\\in \\ma
thbb{Z}$ and $\\mathcal{A}(x)\\gg B(x\\epsilon(x))$. In 2022\, Nathanson f
urther considered Sidon sets for linear forms. Recently\, we apply the Ins
erting Zeros Transformation into Sidon sets for linear forms and generaliz
e the above result related to the inverse problem of representation functi
ons.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/29
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:N. Hegyv\\'ari (E\\"otv\\"os Lor\\' and University and R\\'enyi In
stitute\, Hungary)
DTSTART:20240523T133000Z
DTEND:20240523T135500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/30
DESCRIPTION:Title: On the structures of sets in $\\mathbb{N}^k$ having
thin subset sums\nby N. Hegyv\\'ari (E\\"otv\\"os Lor\\' and Universit
y and R\\'enyi Institute\, Hungary) as part of Combinatorial and additive
number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in th
e CUNY Graduate Center.\n\nAbstract\nFor any $X\\subseteq \\mathbb{N}^k$ l
et\n\\[\nFS(X):=\\{\\sum_{i=1}^\\infty\\varepsilon_ix_i: \\ x_i\\in X\, \\
\\varepsilon_i \\in \\{0\,1\\}\, \\ \\sum_{i=1}^\\infty\\varepsilon_i<\\i
nfty\\}\n\\]\nErdős called a sequence $A\\subseteq \\mathbb{N}$ complete
if every sufficiently large number belongs to $FS(A)$. \nIn a higher dime
nsion too\, the necessary condition that the subset sums of a subset $X\\s
ubseteq \\mathbb{N}^k$ represent all far points of $\\mathbb{N}^k$ should
be the condition $X(N)>k\\log_2N+t_X$ for some $t_X$\, i.e. $X$ is complet
e respect to the region $R=\\{x=(x_1\,x_2\,\\dots\,x_k):x_i\\geq r_i\\}$\,
$r_i\\in \\mathbb{N}$\, $i=1\,2\,\\dots\, k$.\n\n\n We say that $A$ is we
akly thin if $\\limsup_{n\\to \\infty }\\frac{\\log a_n}{\\log n}=\\infty$
\, or equivalently $A(n):= \\sum_{a_i\\leq n}1=n^{g(n)}$\, where $A(n)$ is
the counting function of $A$ and $\\liminf_{n\\to \\infty }g(n)=0$. \nA s
et $B\\subseteq \\mathbb{N}$ is said to be thick if it is not weakly thin.
\nLet $X\\subseteq \\mathbb{N}^k$.\n$X$ is said to be thin complete set re
spect to $R$ if $X(N)>k\\log_2R(N)+t_X$ for some $t_X$ and $FS(X)\\supset
eq R$. \n\nWe prove that if $R=z+\\mathbb{N}^k$ and $A$ is complete with r
espect to $R$\, then all projections of $A$ onto for all axis $f_i$ are th
ick.\nWe also determine the regions for which thin complete sets exist. We
also invetigate the structure of $FS(Y)$\, where $Y=\\{a_m\\}_{m\\in \n}\
\times\\{b_k\\}_{k\\in \n}$ and $\\{a_m\\}_{m\\in \n}$ is a 'dense' sequen
ce.\n\n\n\nThis is a joint work with Máté Pálfy and Erfei Yue.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/30
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sukumar Das Adhikari (Ramakrishna Mission Vivekananda Educational
and Research Institute (RKMVERI)\, India)
DTSTART:20240523T140000Z
DTEND:20240523T142500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/31
DESCRIPTION:Title: Some elementary algebraic and combinatorial methods
in the study of zero-sum theorems\nby Sukumar Das Adhikari (Ramakrishn
a Mission Vivekananda Educational and Research Institute (RKMVERI)\, India
) as part of Combinatorial and additive number theory (CANT 2024)\n\nLectu
re held in Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstrac
t\nOriginating from a beautiful theorem of Erdos-Ginzberg-Ziv about sixty
years ago and some other \nquestions asked around the same time\, the area
of zero-sum theorems has many interesting results \nand several unanswere
d questions.\n\nSeveral authors have introduced interesting elementary alg
ebraic techniques to deal with these problems.\nWe describe some experimen
ts with these elementary algebraic methods and some combinatorial ones\, \
nin a weighted generalization in the area of Zero-sum Combinatorics.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/31
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Leonetti (Università degli Studi dell'Insubria\, Italy)
DTSTART:20240523T143000Z
DTEND:20240523T145500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/32
DESCRIPTION:Title: Most numbers are not normal\nby Paolo Leonetti (
Università degli Studi dell'Insubria\, Italy) as part of Combinatorial an
d additive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room
9207 in the CUNY Graduate Center.\n\nAbstract\nLet $S$ be the set of real
numbers $x \\in (0\,1]$ with the following property of being \\textquoted
blleft strongly not normal": \nFor all integers $b\\ge 2$ and $k\\ge 1$\,
the sequence of vectors made by the frequencies of all possible strings of
length $k$ in the $b$-adic representation of $x$ has a maximal subset of
accumulation points\, and each of them is the limit of a subsequence with
an index set of nonzero asymptotic density. \n\\vskip 0\,05cm\nWe show tha
t $S$ is a co-meager subset of $(0\,1]$\, hence topologically large. Analo
gues are given in the context of regular matrices.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/32
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Dolores Cuenca (Pusan National University\, Korea)
DTSTART:20240523T150000Z
DTEND:20240523T152500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/33
DESCRIPTION:Title: Zeta values as an algebra over an operad\nby Eri
c Dolores Cuenca (Pusan National University\, Korea) as part of Combinator
ial and additive number theory (CANT 2024)\n\nLecture held in Room 4102 an
d Room 9207 in the CUNY Graduate Center.\n\nAbstract\nDenote the operad of
finite posets by FP. In number theory\, the field of rational zeta serie
s studies series of the form $\\sum_{i=1}^\\infty a_i (\\zeta(i+1)-1)\, a_
i\\in\\mathbb{Q}\\\,\\forall i\\in\\mathbb{N}$\, where $\\zeta(k)$ is the
Riemann zeta function $\\zeta(k)=\\sum_{n=1}^\\infty\\frac{1}{n^k}$. By st
udying zeta values as algebras over the operad of posets\,\nwe show the fo
llowing identity\, for $a>1\, a\\in\\mathbb{N}:$\n\n$$\\sum_{n=i}^\\infty
(-1)^{n+1}{n\\choose i}\\zeta(n+1\,a)=(-1)^{i+1}\\zeta(i+1\,a+1)\,$$\nhere
\, $\\zeta(k\,a)=\\sum_{n=0}^\\infty\\frac{1}{(n+a)^k}$ is the Hurwitz zet
a function. \n\nOn January 2023 we put the left side of the identity on se
veral private software\, but none of them produced any output. We presente
d our work in the Wolfram Technology Conference 2023\, where their team ki
ndly verified that the left side of the identity is equal to the right sid
e of the identity. \\\\\nJoint work with Jose Mendoza-Cortes\, Michigan St
ate University\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/33
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qitong (George) Luan (University of California\, Los Angeles)
DTSTART:20240523T153000Z
DTEND:20240523T155500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/34
DESCRIPTION:Title: On a pair of diophantine equations\nby Qitong (G
eorge) Luan (University of California\, Los Angeles) as part of Combinator
ial and additive number theory (CANT 2024)\n\nLecture held in Room 4102 an
d Room 9207 in the CUNY Graduate Center.\n\nAbstract\nFor relatively prime
natural numbers $a$ and $b$\, we study the two equations $ax+by = (a-1)(b
-1)/2$ and $ax+by+1=\n(a-1)(b-1)/2$\, which arise from the study of cyclot
omic polynomials. Previous work showed that exactly one equation has\na no
nnegative integer solution\, and the solution is unique. Our first result
gives criteria to determine which equation\nis used for a given pair $(a\,
b)$. We then use the criteria to study the sequence of equations used by t
he pair\n$(a_n/\\gcd{(a_n\, a_{n+1})}\, a_{n+1}/\\gcd{(a_n\, a_{n+1})})$ f
rom several special sequences $(a_n)_{n\\geq 1}$\, such as\narithmetic pro
gressions\, geometric progressions and sequences satisfying Fibonacci-type
recurrences. Furthermore\, for\neach positive $k$\, we construct a sequen
ce $(a_n)_{n}$ whose consecutive terms use the two equations alternatively
in\ngroups of $k$. Lastly\, we investigate the periodicity of the sequenc
e of equations used by the pair $(k/\\gcd{(k\, n)}\,\nn/\\gcd{(k\, n)})$
as $n$ increases.\\\\\nJoint work with Sujith Uthsara Kalansuriya Arachchi
\, \nH\\`ung Vi\\d{\\^e}t Chu\, Jiasen Liu\, Rukshan Marasinghe\, and Stev
en J. Miller.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/34
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Russell Jay Hendel (Towson University)
DTSTART:20240523T160000Z
DTEND:20240523T162500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/35
DESCRIPTION:Title: Local distance-resistance functions equivalent to gl
obal symmetries in electric circuit families\nby Russell Jay Hendel (T
owson University) as part of Combinatorial and additive number theory (CAN
T 2024)\n\nLecture held in Room 4102 and Room 9207 in the CUNY Graduate Ce
nter.\n\nAbstract\nIn a recent paper Hendel explored the computational att
ributes of an algorithm introduced by Barrett\, Evans\, and Francis\, whic
h\, among other things\, studied distance resistance in a family of circui
ts whose underlying graphs consisted of $n$ rows of upright equilateral tr
iangles ($n$-grids). Two important conjectures supported by numerical evid
ence were presented: one related to the asymptotic behavior of iterated us
e of the algorithm on an initial $n$ grid as $n$ goes to infinity. The se
cond conjecture showed that as $n$ grows large certain limiting ratios eme
rge among specified edges in the circuits resulting from a large number of
repeated applications of the algorithm to an initial $n$-grid. The purpos
e of this paper is to provide insight into these asymptotic or limiting e
dge ratios. After introducing the algorithm and reviewing the original con
jectures\, the main part of this paper studies a family of $n$-grids who
se edge labels are determined using these limiting edge-ratios functions.
The main result proven is that these $n$-grids as well as the graphs
derived from repeated application of the algorithm possess vertical and
rotational symmetries and also continue to satisfy the relationships capt
ured by the limiting edge-ratio functions. In other words\, the limiting e
dge-ratio relationships are local algebraic relationships mirroring the g
lobal vertical and rotational symmetries possessed by the underlying graph
. Additionally\, because row-reduction is local (in contrast to the combin
atoric Laplacian which is global) the paper is able to introduce a mechan
ical verification method of proof for assertions about effective resistanc
e identities.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/35
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin O'Bryant (College of Staten Island (CUNY)\, The Graduate Cen
ter (CUNY))
DTSTART:20240523T173000Z
DTEND:20240523T175500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/36
DESCRIPTION:Title: Greedy $B_h$-sets\nby Kevin O'Bryant (College of
Staten Island (CUNY)\, The Graduate Center (CUNY)) as part of Combinatori
al and additive number theory (CANT 2024)\n\nLecture held in Room 4102 and
Room 9207 in the CUNY Graduate Center.\n\nAbstract\nA set $X$ of integers
is a $B_h$-set if every solution to $\na_1+\\cdots+a_h=b_1+\\cdots +b_h$
with $a_i\,b_i\\in X$ has\n$\\{a_1\,\\ldots\,a_h\\}=\\{b_1\,\\dots\,b_h\\}
$ (as multisets). The main\nproblem is to give inequalities connecting the
cardinality and\ndiameter of $B_h$-sets\, and one obvious way to build th
ick $B_h$-sets\nis to be greedy. In this talk we survey old and new result
s on the\ngreedy $B_h$-sets. The highlight of the new results is a nontriv
ial\nupper bound on the $k$-th element of the greedy $B_h$-set\, provided\
nthat $h$ is sufficiently large.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/36
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Duker Lichtman (Stanford University)
DTSTART:20240523T180000Z
DTEND:20240523T182500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/37
DESCRIPTION:Title: Goldbach beyond the square-root barrier\nby Jare
d Duker Lichtman (Stanford University) as part of Combinatorial and additi
ve number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in
the CUNY Graduate Center.\n\nAbstract\nWe show the primes have level of d
istribution 66/107 using triply well-factorable weights\, and extend this
level to 5/8 assuming Selberg's eigenvalue conjecture. This improves on th
e prior world record level of 3/5 by Maynard. As a result\, we obtain new
upper bounds for Goldbach representations of even numbers. This is the fir
st use of a level of distribution beyond the 'square-root barrier' for the
Goldbach problem\, and leads to the greatest improvement on the problem s
ince Bombieri-Davenport from 1966.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/37
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor D. Wooley (Purdue University)
DTSTART:20240523T183000Z
DTEND:20240523T185500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/38
DESCRIPTION:Title: Unrepresentation theory and sums of powers\nby T
revor D. Wooley (Purdue University) as part of Combinatorial and additive
number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in th
e CUNY Graduate Center.\n\nAbstract\nWe report on recent and on-going work
joint with J\\" org Br\\" udern concerning problems involving the represe
ntation of integer sequences by sums of powers. Our new tool is an upper b
ound for moments of smooth Weyl sums restricted to major arcs. This permit
s progress to be made on Waring's problem and other problems involving mix
ed sums of powers and primes. We will focus on recent progress concerning
unrepresentation theory (bounds for exceptional sets).\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/38
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brad Isaacson (NYC College of Technology (CUNY))
DTSTART:20240523T190000Z
DTEND:20240523T192500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/39
DESCRIPTION:Title: On a reciprocity formula for generalized Dedekind-Ra
demacher sums attached to three Dirichlet characters\nby Brad Isaacson
(NYC College of Technology (CUNY)) as part of Combinatorial and additive
number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 in th
e CUNY Graduate Center.\n\nAbstract\nWe define a three character analogue
of the generalized Dedekind-Rademacher sum introduced by Hall\, Wilson\, a
nd Zagier\, and state its reciprocity formula\, which contains all of the
reciprocity formulas in the literature for generalized Dedekind-Rademacher
sums attached (and not attached) to Dirichlet characters as special cases
. We also review some of the generalized Dedekind-Rademacher sums in the
literature to motivate our results.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/39
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Firdavs Rakhmonov (University of Rochester)
DTSTART:20240523T200000Z
DTEND:20240523T202500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/41
DESCRIPTION:Title: The quotient set of the quadratic distance set over
finite fields\nby Firdavs Rakhmonov (University of Rochester) as part
of Combinatorial and additive number theory (CANT 2024)\n\nLecture held in
Room 4102 and Room 9207 in the CUNY Graduate Center.\n\nAbstract\nLet $\\
mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\
\mathbb F_q$ with $q$ elements. For each non-zero $r$ in $\\mathbb F_q$ an
d $E\\subset \\mathbb F_q^d$\, we define $W(r)$ as the number of quadrupl
es $(x\,y\,z\,w)\\in E^4$ such that $\nQ(x-y)/Q(z-w)=r\,$ where $Q$ is a n
on-degenerate quadratic form in $d$ variables over $\\mathbb F_q.$\nWhen $
Q(\\alpha)=\\sum_{i=1}^d \\alpha_i^2$ with $\\alpha=(\\alpha_1\, \\ldots\,
\\alpha_d)\\in \\mathbb F_q^d\,$ \nPham (2022) recently used the machiner
y of group actions and proved that if $E\\subset \\mathbb F_q^2$ with $q\
\equiv 3 \\pmod{4}$ and $|E|\\ge C q$\, then we have $W(r)\\ge c |E|^4/q$
for any non-zero square number $r \\in \\mathbb F_q\,$ where $C$ is a suf
ficiently large constant\, $ c$ is some number between $0$ and $1\,$ and $
|E|$ denotes the cardinality of the set $E.$\n%In this talk\, \nI'll discu
ss the improvement and extension of Pham's result in two dimensions to arb
itrary dimensions with general non-degenerate quadratic distances. As a c
orollary\, we also generalize the sharp results on the Falconer type probl
em for the quotient set of distance set due to Iosevich-Koh-Parshall. Furt
hermore\, we provide improved constants for the size conditions of the und
erlying sets.\nJoint work with Alex Iosevich and Doowon Koh.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/41
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noah Kravitz (Princeton University)
DTSTART:20240523T203000Z
DTEND:20240523T205500Z
DTSTAMP:20260422T212604Z
UID:NumberTheoryAndCombinatorics/42
DESCRIPTION:Title: Can you reconstruct a set from its subset sums?\
nby Noah Kravitz (Princeton University) as part of Combinatorial and addit
ive number theory (CANT 2024)\n\nLecture held in Room 4102 and Room 9207 i
n the CUNY Graduate Center.\n\nAbstract\nFor a finite subset $A$ of an abe
lian group\, let $\\text{FS}(A)$ denote the multiset of its $2^{|A|}$ subs
et sums. Can you reconstruct $A$ from $\\text{FS}(A)?$ The answer in gener
al is ``no'' (for instance\, $\\text{FS}(\\{1\,3\,-4\\})=\\text{FS}(\\{-1\
,-3\,4\\})$)\, but in many cases\, such as when the ambient group has no $
2$-torsion\, we can obtain a combinatorial description of the fibers of $\
\text{FS}$. \n\nJoint work with Federico Glaudo.\n
LOCATION:https://researchseminars.org/talk/NumberTheoryAndCombinatorics/42
/
END:VEVENT
END:VCALENDAR