CANT 2024.

Twenty-second Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 22 - 24, 2024

Abstracts(1) Sukumar Das Adhikari, Ramakrishna Mission Vivekananda Educational
and Research Institute (RKMVERI), India
Email: adhikarisukumar@gmail.com
Title: Some elementary algebraic and combinatorial methods in the studyof zero-sum theorems
Abstract: Originating from a beautiful theorem of Erdos-Ginzberg-Ziv
about sixty years ago and some other questions asked around the sametime, the area of zero-sum theorems has many interesting results and several unanswered questions.
Several authors have introduced interesting elementary algebraic techniques to deal with these problems. We describe some experiments withthese elementary algebraic methods and some combinatorial ones, in aweighted generalization in the area of Zero-sum Combinatorics.
(2) Adrian Beker, University of Zagreb, Croatia
Email: adrian.beker@math.hr
Title: On a problem of Erdős and Graham about consecutive sums instrictly increasing sequences
Abstract: Given a finite sequence of integers a = (ai )1≤i≤k , letPS(a) denotev
the set of its consecutive sums, that is, sums of the form
i=u ai with1 ≤ u ≤ v ≤ k. Erdős and Graham asked whether there exists a constantc > 0 such that, for all positive integers n, there is such a sequence in{1, . . . , n} which is strictly increasing and satisfies |S(a)| ≥ cn2 .
The obvious candidate consisting of all integers from 1 up to n falls shortof having this property due to reasons related to the multiplication tableproblem. On the other hand, if we drop the monotonicity assumption, suchsequences were shown to exist by Hegyvári via a construction based on
Sidon sets. In this talk, I will present two constructions, one probabilistic
and the other deterministic, that give an affirmative answer to the starting
question. I will also discuss some non-trivial upper bounds on the size ofS(a) in this setting.
(3) Christine K. Chang, CUNY Graduate Center
Email: cchang1@gradcenter.cuny.edu
Title: Hybrid statistics of the maxima of a random model of the zeta function over short intervals
Abstract: We will present a matching upper and lower bound for the right1

tail probability of the maximum of a random model of the Riemann zetafunction over short intervals. In particular, we show that the right tailinterpolates between that of log-correlated and IID random variables as
the interval varies in length. We will also discuss a new normalization
for the moments over short intervals. This result follows the recent workof Arguin-Dubach-Hartung and is inspired by a conjecture by FyodorovHiary-Keating on the local maximum over short intervals.
(4) Jonathan Chapman, University of Bristol, UK
Email: jonathan.chapman@bristol.ac.uk
Title: Monochromatic sums and products
Abstract: If we colour {2, . . . , N } with r different colours, how many
monochromatic solutions to xy = z appear? A classical theorem of Schurshows that we always obtain (cr + o(1))N 2 monochromatic solutions (asN → ∞) to x + y = z, for some cr > 0, which is within a constant factor of
the total number of solutions. However, Prendiville showed that one cannot
achieve such a strong result for xy = z, even if one only uses 2 colours.
In this talk, I will present recent work on determining the asymptotic
minimum number of monochromatic solutions to xy = z. We provethat√
every 2-colouring of {2, . . . , N } produces at least (2−3/2 + o(1)) N log N
monochromatic solutions to xy = z, and the leading constant is sharp. I
will also introduce a Schur-type problem for colourings of real numbers. If
the discrete and continuous Schur problems are ‘quantitatively equivalent’,
then, for an arbitrary number of colours, our upper and lower bounds for thenumber of monochromatic solutions to xy = z match up to a logarithmic
factor.
Joint work with Lucas Aragão, Miquel Ortega, and Victor Souza.
(5) Eric Dolores Cuenca, Pusan National University, Korea
Email: eric.rubiel@pusan.ac.kr
Title: Zeta values as an algebra over an operad
Abstract: Denote the operad of finite posets by FP. In
number theory,P∞
the field of rational zeta series studies series of the form i=1 ai (ζ(iP∞+ 1) −1), ai ∈ Q ∀i ∈ N, where ζ(k) is the Riemann zeta function ζ(k) = n=1 n1k .
By studying zeta values as algebras over the operad of posets, we show thefollowing identity, for a > 1, a ∈ N :
 
∞
X
n+1 n
(−1)
ζ(n + 1, a) = (−1)i+1 ζ(i + 1, a + 1),
i
n=i
P∞
1
here, ζ(k, a) = n=0 (n+a)
k is the Hurwitz zeta function.
On January 2023 we put the left side of the identity on several private
software, but none of them produced any output. We presented our work in
the Wolfram Technology Conference 2023, where their team kindly verified
that the left side of the identity is equal to the right side of the identity.
Joint work with Jose Mendoza-Cortes, Michigan State University

(6) James Cumberbatch, Purdue University
Email: jcumberb@purdue.edu
Title: Digitally restricted sets and the Goldbach conjecture
Abstract: We show that given any base b and any set of digits D with atleast two digits, let A be the set of integers whose base-b digits consist onlyof values in D. We prove that almost all even integers in A are sum of two
primes.
(7) Robert Donley, Queensborough Community College (CUNY)
Email: rdonley@qcc.cuny.edu
Title: A combinatorial introduction to adinkras
Abstract: In 2005, Faux and Gates defined the adinkra, a graphical devicefor describing particle exchanges in supersymmetry. Independent of thephysical applications, the adinkra 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.
Joint work with S. James Gates, Jr., Tristan Hübsch, and Rishi Nath.
(8) Jin-Hui Fang, Nanjing Normal University, China
Email: fangjinhui1114@163.com
Title: On Cilleruelo-Nathanson’s method in Sidon sets
Abstract: For nonnegative integers h, g with h ≥ 2, a set A of nonnegative
integers is defined as a Bh [g] sequence if, for every nonnegative integer n,
the number of representations of n with the form n = a1 + a2 + · · · + ah
is no larger than g, where a1 ≤ · · · ≤ ah and ai ∈ A for i = 1, 2, · · · , h.
Let Z be the set of integers and N be the set of positive integers. In 2013,
by introducing the method of Inserting Zeros Transformation, Cilleruelo
S
and Nathanson obtained the following nice result: let f : Z → N {0, ∞}be any function such that lim inf |n|→∞ f (n) ≥ g and let B be any Bh [g]sequence. Then, for any decreasing function (x) → 0 as x → ∞, there
exists a sequence A of integers such that rA,h (n) = f (n) for all n ∈ Z andA(x)  B(x(x)). In 2022, Nathanson further considered Sidon sets forlinear forms. Recently, we apply the Inserting Zeros Transformation into
Sidon sets for linear forms and generalize the above result related to theinverse problem of representation functions.
(9) Leonid Fel, Technion – Israel Institute of Technology, Israel
Email: lfel@cv.technion.ac.il
Title: Ratio between a sum of generators and rational powers of their product√
Pm
Abstract: We study a ratio Rm (k) = I1 / k Im between a sum I1 = j=1 dj√Qmof generators and rational powers k Im of their product Im = j=1 dj in+numerical semigroups hd1 , . . . , dm i. We find its upper Rm (k) and lowerR−m (k) bounds in the range 1 ≤ k ≤ m. We prove that Rm (k) has a universal upper bound if and only if m ≥ 2k − 1.

(10) Daniel Flores, Purdue University
Email: flore205@purdue.edu
Title: A circle method approach to K-multimagic squares
Abstract: In this paper we investigate K-multimagic squares of order N ,which are N × N magic squares with remain magic after raising each element to the kth power for all 2 ≤ k ≤ K. Given K ≥ 2, we consider the
problem of establishing the smallest integer N (K) for which there exists
non-trivial K-multimagic squares of order N (K). Previous results on multimagic squares show that N (K) ≤ (4K − 2)K for large K. Here we utilize
the Hardy-Littlewood circle method and establish the bound
N (K) ≤ 2K(K + 1) + 1.
Via an argument of Granville’s we additionally deduce the existence of
infinitely many non-trivial prime valued K-multimagic squares of order
2K(K + 1) + 1.
(11) Krystian Gajdzica, Jagiellonian University, Kraków, Poland
Email: krystian.gajdzica@im.uj.edu.pl
Title: A combinatorial approach to the Bessenrodt-Ono type inequalities
Abstract: In 2016, Bessenrodt and Ono showed that the partition function
satisfies the inequality of the form
p(a)p(b) > p(a + b)
for all a, b > 2 with a + b > 9. Their proof is based on the asymptotic
estimates of p(n) due to Lehmer. Since then, a lot of similar phenomena
have been discovered for various variations of the partition function.
We discuss the analogue of the Bessenrodt-Ono inequality for the socalled A-partition function pA (n), which enumerates those partitions of nwhose parts belong to a fixed set A ⊂ N. Since there is no known asymptotic formula for pA (n) in general, we can not deal with the problem using
any estimates of pA (n). Therefore, we present a combinatorial approachto the issue by constructing an appropriate injection between some sets ofpartitions.
(12) N. Hegyvári, Eötvös Loránd University and Rényi Institute, Hungary
Email: hegyvari@renyi.hu
Title: On the structures of sets in Nk having thin subset sums
Abstract: For any X ⊆ Nk let∞∞
X
X
F S(X) := {εi xi : xi ∈ X, εi ∈ {0, 1},εi < ∞}
i=1

i=1

Erdős called a sequence A ⊆ N complete if every sufficiently large number
belongs to F S(A). In a higher dimension too, the necessary condition thatthe subset sums of a subset X ⊆ Nk represent all far points of Nk should be
the condition X(N ) > k log2 N − tX for some tX , i.e. X is complete respect
to the region R = {x = (x1 , x2 , . . . , xk ) : xi ≥ ri }, ri ∈ N, i = 1, 2, . . . , k.
Let A = {a1 < a2 < · · · < an < . . . } be an infinite sequence of integers.
an
We say that A is weakly thin if lim supn→∞ log
log n = ∞, or equivalentlyP
A(n) := ai ≤n 1 = ng(n) , where A(n) is the counting function of A andlim inf n→∞ g(n) = 0. A set B ⊆ N is said to be thick if it is not weakly
thin. Let X ⊆ Nk . X is said to be thin complete set respect to R if
X(N ) > k log2 R(N ) − tX for some tX and F S(X) ⊇ R. We prove that if
R = z + Nk and A is complete with respect to R, then all projections ofA onto for all axis fi are thick. We also determine regions for which there
exists thin complete sets. Some related results are also discussed.
This is joint work with Máté Pálfy and Erfei Yue.
(13) Russell Jay Hendel, Towson University
Email: RHendel@Towson.edu
Title: Local distance-resistance functions equivalent to global symmetries
in electric circuit families
Abstract: In a recent paper Hendel explored the computational attributesof an algorithm introduced by Barrett, Evans, and Francis, which, among
other things, studied distance resistance in a family of circuits whose underlying graphs consisted of n rows of upright equilateral triangles (ngrids). Two important conjectures supported by numerical evidence were
presented: one related to the asymptotic behavior of iterated use of thealgorithm on an initial n grid as n goes to infinity. The second conjecture
showed that as n grows large certain limiting ratios emerge among specifiededges in the circuits resulting from a large number of repeated applicationsof the algorithm to an initial n-grid. The purpose of this paper is to provide insight into these asymptotic or limiting edge ratios. After introducing
the algorithm and reviewing the original conjectures, the main part of thispaper studies a family of n-grids whose edge labels are determined using
these limiting edge-ratios functions. The main result proven is that thesen-grids as well as the graphs derived from repeated application of the algorithm possess vertical and rotational symmetries and also continue tosatisfy the relationships captured by the limiting edge-ratio functions. In
other words, the limiting edge-ratio relationships are local algebraic relationships mirroring the global vertical and rotational symmetries possessedby the underlying graph. Additionally, because row-reduction is local (incontrast to the combinatoric Laplacian which is global) the paper is able
to introduce a mechanical verification method of proof for assertions about
effective resistance identities.
(14) Brad Isaacson, NYC College of Technology (CUNY)
Email: brad.isaacson71@citytech.cuny.edu
Title: On a reciprocity formula for generalized Dedekind-Rademacher sums
attached to three Dirichlet characters
Abstract: We define a three character analogue of the generalized DedekindRademacher sum introduced by Hall, Wilson, and Zagier, and state its
reciprocity formula, which contains all of the reciprocity formulas in theliterature 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.

(15) Rauan Kaldybayev, Williams College
Email: rk19@williams.edu
Title: Limiting behavior in missing sums of sumsets
Abstract: We study |A + A| as a random variable, where A ⊆ {0, . . . , N }is a random subset such that each 0 ≤ n ≤ N is included with probability
0 < p < 1, and where A + A is the set of sums a + b for a, b in A. Lazarev,
Miller, and O’Bryant studied the distribution of 2N + 1 − |A + A|, the number of summands not represented in A+A when p = 1/2. A recent paper by
Chu, King, Luntzlara, Martinez, Miller, Shao, Sun, and Xu generalizes thisto all p ∈ (0, 1), calculating the first and second moments of the number ofmissing summands and establishing exponential upper and lower bounds on
the probability of missing exactly n summands, mostly working in the limitof large N . We provide exponential bounds on the probability of missing
at least n summands, find another expression for the second moment of
the number of missing summands, extract its leading-order behavior in thelimit of small p, and show that the variance grows asymptotically slower
than the mean, proving that for small p, the number of missing summands
is very likely to be near its expected value.
(16) Mizan Khan, Eastern Connecticut State University
Email: khanm@easternct.edu
Title: A conjecture for clean lattice parallelograms
Abstract: Let P ⊆ R2 be a convex lattice polygon containing at least one
lattice point in its interior. The interior hull of P , denoted by P (1) , is theconvex closure of the set of lattice points in the interior of P , that is,
P (1) = conv interior(P ) ∩ Z2 .
We can now form a finite nested sequence of interior hulls
P (1) ⊇ P (2) ⊇ P (3) ⊇ . . . ,where P (2) is the interior hull of P (1) , P (3) is the interior hull of P (2) , andso on.
We will present some experimental data supporting a conjecture on the
average number of interior hulls for clean lattice parallelograms. (A lattice
parallelogram is said to be clean if the only lattice points on its boundary
are the 4 vertices.)
Joint work with Riaz Khan.
(17) Nathaniel Kingsbury, CUNY Graduate Center
Email: nkingsbury@gradcenter.cuny.edu
Title: The square-root law does not hold in the presence of zero divisors
Abstract: Let R be a finite ring and define the paraboloid P = {(x1 , . . . , xd ) ∈Rd |xd = x21 + · · · + x2d−1 }. Suppose that for a sequence of finite rings of size
tending to infinity, the Fourier transform of P satisfies a square-root typebound constant C. Then all but finitely many of the rings are fields.
Most of our argument works in greater generality: let f be a polynomialwith integer coefficients in d − 1 variables, with a fixed order of variablemultiplications (so that it defines a function Rd−1 → R even when R is
noncommutative), and set Vf = {(x1 , . . . , xd ) ∈ Rd |xd = f (x1 , . . . , xd−1 )}.

If (for a sequence of finite rings of size tending to infinity) we have a squareroot type bound on the Fourier transform of Vf , then all but finitely manyof the rings are fields or matrix rings of small dimension.
(18) Gergely Kiss, Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Email: kigergo57@gmail.com
Title: Solutions to the discrete Pompeiu problem and to the finite Steinhaus
tiling problem
Abstract: Let K be a nonempty finite subset of the Euclidean space Rk(k ≥ 2). In this talk we discuss the solution of the following so-called
discrete Pompeiu problem. If a function f : Rk → C is such that the sumof f on every congruent copy of K is zero, then f vanishes everywhere. In
fact, we solve a stronger, weighted version of this problem. As a corollary
we obtain that every finite subset of Rk having at least two elements is a
Jackson set; that is, no subset of Rk intersects every congruent copy of K
in exactly one point.
Joint work with Miklós Laczkovich.
(19) Sergei Konyagin, Steklov Institute of Mathematics, Russia
Email: konyagin@mi-ras.ru
Title: On distinct angles in the plane
Abstract: The talk is based on our joint paper with Jonathan Passant andMisha Rudnev. We prove that if N points lie in convex position in theplane, then they determine  N 1+3/23+o(1) distinct angles, provided noN − 1 points lie on a common circle. This is the first super–linear boundon the distinct angle problem that has received recent attention.
(20) Noah Kravitz, Princeton University
Email: nkravitz@princeton.edu
Title: Can you reconstruct a set from its subset sums?
Abstract: For a finite subset A of an abelian group, let FS(A) denote themultiset of its 2|A| subset sums. Can you reconstruct A from FS(A)? The
answer in general is “no” (for instance, FS({1, 3, −4}) = 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 FS.
Joint work with Federico Glaudo.
(21) Andrzej Kukla, Jagiellonian University in Kraków, Poland
Email: andrzej.kukla@student.uj.edu.pl
Title: Representing the number of binary partitions as sums of three squares
Abstract: Let cm (n) denote the number of partitions of n into parts thatare powers of 2 such that part equal to 1 takes one among 2m colors andeach part > 1 takes one among m colors. The study of this function was
initiated in 2021 by Zmija and Ulas, who focused on its 2-adic behaviour.
In particular, they found a formula for the 2-adic valuation of cm (n) that
depends on the 2-adic valuation of m and the value of tn − tn−1 , where tn
is the n-th term of the Prouhet-Thue-Morse sequence. During the talk wewill be considering the diophantine equation cm (n) = x2 +y 2 +z 2 . For fixed

m, we will characterize the set of natural numbers n, for which the solution
does not exist, and then further investigate properties of these sets. The
talk is based on an ongoing work on speaker’s master’s thesis.
(22) Noah Lebowitz-Lockard, University of Texas, Tyler, TX
Email: nlebowitzlockard@uttyler.edu
Title: On the smallest parts of partitions into distinct parts
Abstract: For a given integer n, let D(n) be the set of partitions of n into
distinct parts. Create a sum as follows. For each partition λ in D(n), add
the smallest element of λ if it is even and subtract it if it is odd. A classic theorem of Uchimura states that this quantity is equal to the numberof divisors of n. We generalize this result to the sum of the kth smallest
elements of partitions for a fixed value of k. We also consider some further
generalizations, as well as variants for the smallest number not in a given
partition.
Joint work with Rajat Gupta and Joseph Vandehey.
(23) Paolo Leonetti, Universitá degli Studi dell’Insubria, Italy
Email: leonettipaolo@gmail.com
Title: Most numbers are not normal
Abstract: Let S be the set of real numbers x ∈ (0, 1] with the following
property of being “strongly not normal”: For all integers b ≥ 2 and k ≥ 1,
the sequence of vectors made by the frequencies of all possible strings oflength 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.
We show that S is a co-meager subset of (0, 1], hence topologically large.
Analogues are given in the context of regular matrices.
(24) Jared Duker Lichtman, Stanford University
Email: jared.d.lichtman@gmail.com
Title: Goldbach beyond the square-root barrier
Abstract: We show the primes have level of distribution 66/107 using triply
well-factorable weights, and extend this level to 5/8 assuming Selberg’s
eigenvalue conjecture. This improves on the 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 first use of a level of distribution
beyond the ‘square-root barrier’ for the Goldbach problem, and leads to thegreatest improvement on the problem since Bombieri-Davenport from 1966.
(25) Qitong (George) Luan, University of California, Los Angeles
Email: qluan21@g.ucla.edu
Title: On a pair of diophantine equations
Abstract: For relatively prime natural numbers a and b, we study the two
equations ax+by = (a−1)(b−1)/2 and ax+by +1 = (a−1)(b−1)/2, whicharise from the study of cyclotomic polynomials. Previous work showed thatexactly one equation has a nonnegative integer solution, and the solution is
unique. Our first result gives criteria to determine which equation is usedfor a given pair (a, b). We then use the criteria to study the sequence ofequations used by the pair (an / gcd (an , an+1 ), an+1 / gcd (an , an+1 )) fromseveral special sequences (an )n≥1 , such as arithmetic progressions, geometric progressions and sequences satisfying Fibonacci-type recurrences.
Furthermore, for each positive k, we construct a sequence (an )n whose consecutive terms use the two equations alternatively in groups of k. Lastly,
we investigate the periodicity of the sequence of equations used by the pair(k/ gcd (k, n), n/ gcd (k, n)) as n increases.
Joint work with Sujith Uthsara Kalansuriya Arachchi, Hùng Viê.t Chu, Jiasen Liu, Rukshan Marasinghe, and Steven J. Miller.
(26) Steve Miller, Williams College
Email: sjm1@williams.edu
Title: The theory of normalization constants and Zeckendorf decompositions
Abstract: If we define the Fibonacci numbers to start 1, 2, 3, 5 and so on, we
have a wonderful property: Every positive integer has a unique representation as a sum of non-adjacent terms. Called the Zeckendorf decomposition,
we can prove many results about the summands, from the number in a
typical decomposition converging to a Gaussian to the probabilities of gaps
converging to a geometric decay. Many of these proofs are straightforward
but tedious exercises in algebra. We present a new approach, which so far
has just been applied to the distribution of gaps, but hopefully can workfor related problems, which bypasses these calculations through the theoryof normalization constants.
(27) Piotr Miska, Jagiellonian University in Kraków, Poland
Email: piotr.miska@uj.edu.pl
Title: On the Frobenius problem with restrictions on common divisors ofcoefficients
Abstract: Let m, s, t be positive integers with t ≤ s−2 and let a1 , a2 , . . . , as
be positive integers such that (a1 , a2 , . . . , as−1 ) = 1. In the paper we provethat every sufficiently large positive integer can be written in the form
a1 µ1 + a2 µ2 + . . . + as µm , where the positive integers µ1 , µ2 , . . . , µs have nocommon divisor that is the m-th power of a positive integer greater than 1,
but each t of the values µ1 , µ2 , . . . , µs do have a common divisor that is them-th power of a positive integer greater than 1. Moreover, we show thatevery sufficiently large positive integer can be written as a sum of positive
integers µ1 , µ2 , . . . , µs with no common divisor that is the m-th power of apositive integer greater than 1, but each s − 1 of the values of µ1 , µ2 , . . . , µs
do have a common divisor that is the m-th power of a positive integer
greater than 1.
Joint work with Maciej Zakarczemny (Cracow University of Technology).

(28) Mel Nathanson, Lehman College (CUNY)
Email: melvyn.nathanson@lehman.cuny.edu
Title: Polynomial equations in infinitely many variables(29) Kevin O’Bryant, College of Staten Island (CUNY)
Email: obryant@gmail.com
Title: Greedy Bh -sets
Abstract: A set X of integers is a Bh -set if every solution to a1 + · · · + ah =b1 + · · · + bh with ai , bi ∈ X has {a1 , . . . , ah } = {b1 , . . . , bh } (as multisets).
The main problem is to give inequalities connecting the cardinality anddiameter of Bh -sets, and one obvious way to build thick Bh -sets is to be
greedy. In this talk we survey old and new results on the greedy Bh -sets.
The highlight of the new results is a nontrivial upper bound on the k-th
element of the greedy Bh -set, provided that h is sufficiently large.
(30) Carl Pomerance, Dartmouth College
Email: carlp@math.dartmouth.edu
Title: Matchable numbers
Abstract: For a natural number n let D(n) denote the set of positive
divisors of n and let τ (n) = #D(n). Say n is matchable if there is abijection from D(n) to {1, 2, . . . , τ (n)} with corresponding numbers relatively prime. For example, each number up to 7 is matchable, but 8 isnot. This definition was made by Santos on MathOverflow in 2022; heasks if there are more matchable numbers than not. We prove this byshowing
the set of matchable numbers has an asymptotic density given byQpp prime (1 − 1/p ) = .72199 . . . .
This is joint work with Nathan McNew.
(31) Firdavs Rakhmonov, University of Rochester
Email: frakhmon@ur.rochester.edu
Title: The quotient set of the quadratic distance set over finite fields
Abstract: Let Fdq be the d-dimensional vector space over the finite field Fq
with q elements. For each non-zero r in Fq and E ⊂ Fdq , we define W (r) as
the number of quadruples (x, y, z, w) ∈ E 4 such that Q(x−y)/Q(z −w) = r,
where Q is a non-degenerate quadratic form in d variables over Fq . WhenPd
Q(α) = i=1 αi2 with α = (α1 , . . . , αd ) ∈ Fdq , Pham (2022) recently used
the machinery of group actions and proved that if E ⊂ F2q with q ≡ 3(mod 4) and |E| ≥ Cq, then we have W (r) ≥ c|E|4 /q for any non-zerosquare number r ∈ Fq , where C is a sufficiently large constant, c is somenumber between 0 and 1, and |E| denotes the cardinality of the set E. I’ll
discuss the improvement and extension of Pham’s result in two dimensions
to arbitrary dimensions with general non-degenerate quadratic distances.
As a corollary, we also generalize the sharp results on the Falconer typeproblem for the quotient set of distance set due to Iosevich-Koh-Parshall.
Furthermore, we provide improved constants for the size conditions of theunderlying sets. Joint work with Alex Iosevich and Doowon Koh.

(32) James Sellers, University of Minnesota Duluth
Email: jsellers@d.umn.edu
Title: Elementary proofs of congruences for POND and PEND partitions
Abstract: Recently, Ballantine and Welch considered two classes of integer
partitions which they labeled POND and PEND partitions. These are integer partitions wherein the odd parts (respectively, the even parts) cannot
be distinct. In recent work, I studied these two types of partitions from anarithmetic perspective and proved infinite families of mod 3 congruences
satisfied by the two corresponding enumerating functions. I will talk about
the generating functions for these enumerating functions, and I will also
highlight the (induction) proofs that I utilized.
(33) Steven Senger, Missouri State University
Email: stevensenger@gmail.com
Title: Multi-parameter point-line incidence estimates in finite fields andapplications
Abstract: We 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 products type problems.
(34) I.D. Shkredov, Purdue University
Email: ilya.shkredov@gmail.com
Title: On universal sets and sumsets
Abstract: Let G be an abelian group. A set A ⊆ G is called a k–universal set if for any x1 , . . . , xk ∈ G there exists s ∈ G such thatx1 + s, . . . , xk + s ∈ A. The term “universal set” was introduced by Alon,
Bukh, and Sudakov in connection with the discrete Kakeya problem. We
study the concept of universal sets from the additive–combinatorial pointof view. Among other results we obtain some applications of this type of
uniformity to sets avoiding solutions to linear equations, and get an optimal
upper bound for the covering number of general sumsets.
(35) Bartosz Sobolewski, Jagiellonian University, Kraków, Poland and Montanuniversität Leoben, Austria
Email: bartosz.sobolewski@uj.edu.pl
Title: On block occurrences in the binary expansions of n and n + t
Abstract: Let s(n) denote the sum of binary digits of a nonnegative integer
n. In the recent years there has been significant progress concerning the behavior of the differences s(n+t)−s(n), where t is a fixed nonnegative integer.
In particular, Spiegelhofer and Wallner proved that for t having sufficiently
many blocks 01 in its binary expansion, the set {n : s(n + t) ≥ s(n)} has
natural density > 1/2 (partially confirming a conjecture by Cusick). Moreover, for such t the distribution s(n + t) − s(n) is close to Gaussian. During
the talk we consider an analogue of this problem concerning the functionr(n), which counts the occurrences of the block 11 in the binary expansionof n. In particular, we prove that the distribution of r(n + t) − r(n) isapproximately Gaussian as well. We also discuss a generalization to anarbitrary block of binary digits.
Joint work with Lukas Spiegelhofer (Montanuniversität Leoben).
(36) Gábor Somlai, Eötvös Loránd University and Rényi Institute, Hungary
Email: zsomlei@gmail.com
Title: New method for old results of Rédei, Lovász and Schrijver
Abstract: Rédei proved that a set S of cardinality p in F2p determines at
directions or S is a line. We managed find a short proof forleast p+3
2
Rédei’s result avoiding the theory of lacunary polynomials by proving thefollowing statement. Let f be a polynomial over the finite field Fp . Consider. . . , p − 1}. Assume that
P the elements of the range as integers in {0, 1,p−1
f
(x)
=
p.
Then
either
f
=
1
or
deg(f
)
≥
x∈Fp
2 . The uniqueness (upto affine transformations) of the sets of size p in F2p was proved by Lovász
and Schrijver. The same result follows from the almost uniqueness of thepolynomials of degree p−1
2 of range sum p.
(37) Christian Táfula, Université de Montréal, Canada
Email: christian.tafula.santos@umontreal.ca
Title: Representation functions with prescribed rates of growth
Abstract: Let h ≥ 2, and b1 , . . ., bh be positive integers with gcd = 1. Fora set A ⊆ N, denote by rA (n) the number of solutions to the equation
b1 k1 + ... + bh kh = n
with k1 , . . ., kh ∈ A. For which functions F can we find A such thatrA (n) ∼ F (n)? Or rA (n)  F (n)? In the asymptotic case, we show thatfor every F “regularly varying” in the range
1
ω(log x) = F (x) ≤ (h−1)!b
+
o(1)
xh−1
1 ...bh(where f = ω(g) if g = o(f )), there is A such that rA (n) ∼ F (n). In theorder of magnitude case, there is A with rA (n)  F (x) for every F nondecreasing such that F (2x)  F (x) in the range log x  F (x)  xh−1 .
This extends earlier work of Erdős–Tetali and Vu, and addresses a question
raised by Nathanson on which functions can be the rA of some A.
If time allows, we will show for 1  F (x) = o(log x) a probabilistic
heuristic for why there should not exist A with rA (n)  F (n), agreeing
with a conjecture of Erdős.
(38) Marc Technau, Paderborn University, Germany
Email: mtechnau@math.tugraz.at
Title: Cilleruelo’s conjecture on the LCM of polynomial sequences
Abstract: We discuss a conjecture of Cilleruelo on the growth of the least
common multiple of consecutive values of a polynomial and subsequent
progress towards it in work of Maynard–Rudnick and Sah. In recent work,
the speaker and, independently, Alexei Entin made further advances by exploiting symmetries amongst the roots of the polynomials in question. We
shall discuss these approaches and related beautiful work of Baier and Dey.

(39) Fred Tyrell, University of Bristol, UK
Email: fred.tyrrell@bristol.ac.uk
Title: New lower bounds for cap sets
Abstract: A cap set is a subset of Fn3 with 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 introduce the problem, givesome background and motivation, and describe how I was able to provide
the first progress in 20 years on the lower bound for the size of a maximal
cap set. Building on a construction of Edel, we use improved computational
methods and new theoretical ideas to show that, for large enough n, there isalways a cap set in Fn3 of size at least 2.218n . I will then also discuss recent
developments, including an extension of this result by Google DeepMind.
(40) Maciej Ulas, Jagiellonian University, Kraków, Poland
Email: maciej.ulas@gmail.com
Title: On the Diophantine equation x3 ± y 3 = ak ± bk
Abstract: We consider the Diophantine equation x3 ± y 3 = ak ± bk fromboth the theoretical as well as the experimental point of view. In particular, we prove that for k = 4, 6, regardless of the signs chosen, our equation
has infinitely many co-prime positive integer solutions. This confirms expectations outlined by Wagstaff. Additionally, for k = 5, 7 we report on
computations of all co-prime positive integer solutions (x, y, a, b) satisfying
the condition max{a, b} ≤ 50000.
(41) Aleksei Volostnov, Moscow Institute of Physics and Technology, Russia
Email: aleksey.volostnov@phystech.edu
Title: On the additive energy of roots
Abstract: Let p be a prime number, f ∈ Fp [x] be a polynomial of small
degree and a set A ⊂ Fp have sufficiently small cardinality in terms of p.
We study the number of solutions to the equation (in Fp )x1 + x2 = x3 + x4 ,

f (x1 ), f (x2 ), f (x3 ), f (x4 ) ∈ A,

provided that A has small doubling. Namely, we improve the upper boundfrom recent work by B. Kerr, I. D. Shkredov, I. E. Shparlinski and A.
Zaharescu.
Moreover, we address questions of cardinalities |A + A| vs |f (A) + f (A)|.
In particular, we prove thatmax(|A + A|, |A3 + A3 |)  |A|16/15
max(|A + A|, |A4 + A4 |)  |A|25/24
max(|A + A|, |A5 + A5 |)  |A|25/24 .

(42) Trevor D. Wooley, Purdue University
Email: twooley@purdue.edu
Title: Unrepresentation theory and sums of powers
Abstract: We report on recent and on-going work joint with Jörg Brüdernconcerning problems involving the representation of integer sequences bysums of powers. Our new tool is an upper bound for moments of smooth
Weyl sums restricted to major arcs. This permits progress to be made on
Waring’s problem and other problems involving mixed sums of powers andprimes. We will focus on recent progress concerning unrepresentation theory (bounds for exceptional sets).