CANT 2019 Abstracts
Seventeenth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 21–24, 2019
Paul Baginski, Fairfield University
Title: Abundant numbers and nonunique factorization
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Abstract: An integer n is abundant if the sum of its divisors, σ(n) = d|n d,
is greater than 2n; n is perfect if σ(n) = 2n; and otherwise n is deficient. Boththe set H of abundant numbers and the set H ∗ of non-deficient numbers are closed
under multiplication, making them submonoids of (N, ×). Both of these monoids
have elements with nonunique factorization into irreducibles. We will discuss thefactorization properties these monoids possess and connections to known problems
about abundant numbers.
Joint work with Jill Stifano.
Amanda Burcroff, University of Michigan
Title: Covering problems involving the unitary Cayley graph of Z/nZ
Abstract: Let Cn denote the set of integers relatively prime to n, and let γt (n)denote the minimum number of translates of Cn needed to cover Z. We answer aquestion of Defant and Iyer by constructing a family of infinitely many integers nsuch that γt (n) ≤ g(n) − 2, where g(n) is the Jacobsthal function, the minimum msuch that every set of m consecutive integers contains an integer coprime to n. The
unitary Cayley graph of Z/nZ, denoted Xn , is the graph on {0, . . . , n − 1} wherevertices a and b are adjacent if a − b is coprime to n. The parameter γt (n) can beviewed graph-theoretically as the total domination number of Xn .
When n is squarefree, we bound the minimum number of translates of Cn ∪ {0}needed to cover Z and the minimum number of translates of Cn by elements of
Cn needed to cover Z. These correspond to the domination number and lower
independence number of Xn , respectively. When n has a bounded smallest prime
factor, we also give asymptotic bounds for the maximum number of translates of Cn
minimally covering an arbitrary subset of Z, i.e., the upper irredundance numberof Xn .
Colin Defant, Princeton University
Title: Connected components of ranges of divisor functions
Abstract: For each complex number c, the divisor function σc is defined by σc (n) =P
cd|n d . This talk considers the topological properties of σc (N), the closure of therange of σc . More precisely, we will discuss N (c), the number of connected components of σc (N). There have been several recent developments toward understanding
N (c) when c is a negative real number. In this setting, Sanna proved that N (c) is
finite, and the combined efforts of Zubrilina, Achenjang, and Berger have yielded1asymptotic estimates for these numbers. We will also see that N (c) is finite whenever c is a complex number with negative real part. The second half of the talkconcerns some wide open problems, including the following question. For whichbounded multiplicative functions f : N → C is it the case that f (N) has finitely
many connected components?
Robert W. Donley, Jr., Queensborough Community College (CUNY)
Title: The gentle art of 3 × 3 semi-magic squares
Abstract: In 1958, the physicist Tullio Regge made the remarkable observation thatthe domain space for electron coupling calculations (Clebsch-Gordan coefficients)
consists of 3 × 3 semi-magic squares and that such computations transform under
the full group of 72 determinantal symmetries on 3×3 matrices. Here “semi-magic”
means that entries are nonnegative integers and that sums along rows and columns
are constant. To visualize the five-dimensional domain space, we propose a threedimensional model whose points represent hexagonal families of semi-magic squares.
In turn, the vanishing points of the Clebsch-Gordan function produce portraits inthis model with intriguing artistic qualities.
Heidi Goodson, Brooklyn College (CUNY)
Title: Vertically aligned entries in Pascal’s triangle and applications to number
theoryAbstract: The classic way to write down Pascal’s triangle leads to entries in alternating rows being vertically aligned. In this talk, I will give a linear dependence onvertically aligned entries in Pascal’s triangle. Furthermore, I will give an application of this dependence to number theory. Specifically, I will explain how a search
for morphisms between hyperelliptic curves led to the discovery of this identity.
David Grynkiewicz, University of Memphis
Title: An inverse zero-sum problem for elementary abelian p-groups of rank two
Abstract: Let G = Z/pZ × Z/pZ be an elementary abelian p-group of rank two.
For an integer k ≥ 1, let s≤k (G) denote the minimal integer ` ≥ 1 such that any
sequence of terms from G having length at least ` must contain a nonempty zerosum subsequence of length at most k. It is trivial that sk (G) = ∞ for k < p, andthat s≤k (G) = s≤D(G) (G) for any k ≥ D(G), where D(G) = 2p − 1 denotes the
Davenport constant of G, which is the minimal integer ` such that any sequence oflength ` must contain a nonempty zero-sum subsequence of some length. The range
of interest is then k ∈ [p, 2p − 1]. The value of s≤k (G) for the extremal values k = p
and k = 2p − 1 are classical, and go back to the work of Olson. More recent workof Wang and Zhao has determined that s≤k (G) = 4p − 2 − k for any k ∈ [p, 2p − 1].
In this talk, we are concerned with the related inverse problem for s≤k (G).
Namely, we wish to completely characterize all sequences S of maximal possible
length s≤k (G) − 1 = 4p − 3 − k that fail to have the desired nonempty zero-sum
subsequence of length at most k. When k = 2p − 2, the problem reduces to characterizing all zero-sum sequences of length 2p − 1 having no proper, nonempty
zero-sum subsequence. A major achievement of Reiher established the conjectured
structure of such sequences (known in the literature as Property B) for all primes

p ≥ 2. The case k = 2p − 1 follows directly from the case k = 2p − 2, and thecharacterization for k = p, known in the literature as Property C, had also beenshown by a more involved argument (of Gao and Geroldinger) to reduce to the casek = 2p − 2, thus following from the work of Reiher as well.
In this talk, we present work characterizing those extremal sequences S when klies in the broader range k ∈ [ 34 (p − 1), 2(p − 1)]. Specifically, we show that if S is asequence of 4p − 3 − k terms from G containing no nonempty zero-sum subsequenceof length at most k, then there must some basis (e1 , e2 ) of G such that S consistsof the terms e1 and e2 , each with multiplicity p − 1, as well as the term e1 + e2 withmultiplicity p − 1 − k. In other words, if we write the sequence S as a multiplicative
string using g [y] to denote the term g ∈ G repeated y ≥ 0 times, S must have thefollowing form:[p−1]

S = e1

[p−1]

· e2

· (e1 + e2 )[p−1−k] .

In contrast to the extremal values k ≥ 2p − 2 and k = p, this shows more regularstructure holds outside the extremal values for k.
Joint work with Chunlin Wang and Kevin Zhao.
Matthew Hase-Liu, Harvard University
Title: Sum-product phenomena for planar hypercomplex numbers
Abstract: We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar tothe complex numbers, but it turns out that they have a very different combinatorial
behavior. We identify parameters that control the behavior of these problems, andderive sum-product bounds that depend on these parameters. For the dual numbers
we expose a range where the minimum value of max{|A + A|, |AA|} is neither close
to |A| nor to |A|2 . To this end, we study point-line incidences in the dual plane
and in the double plane, developing analogs of the Szemerédi-Trotter theorem andextensions of Elekes’ sum-product technique. As in the case of the sum-productproblem, it turns out that the dual and double variants behave differently than thecomplex and real ones.
Charles Helou, Penn State, Brandywine
Title: A characterization of representation functions
Abstract: Given a set A of natural numbers, i.e., nonnegative integers, the representation function of A is the function which to every natural number n associates thenumber rA (n) of ordered pairs (a, b) of elements a, b ∈ A such that a + b = n. We
establish an intrinsic characterization of representation functions, thus answering
an open problem stated many years ago by Mel Nathanson.
Robert Hough, SUNY - Stony Brook Title: Cut-off phenomenon for the abelian
sandpile model on tiling graphs
Abstract: In the abelian sandpile model on a graph G = (V, E) with sink s, there area non-negative number of chips σ(v) at each non-sink vertex v. If σ(v) ≥ deg(v),the vertex can ‘topple’ passing one chip to each neighbor. Any chips which fallon the sink are lost from the model. A configuration σ is called ‘stable’ if novertex can topple. In driven dynamics in the model, at each step a chip is added

to the model at a uniform random vertex, and all legal topplings are performed
until a stable configuration is reached. Together with Dan Jerison and Lionel
Levine, I determined the asymptotic mixing time to stationarity and proved a cutoff phenomenon for dynamics on a square N × N grid with periodic boundary
conditions and a single sink in the limit N → ∞. Recently, with Hyojeong Son, I
have extended this result to prove a cut-off phenomenon for sandpile dynamics on a
growing piece of an arbitrary plane or space tiling, with open or periodic boundary
condition, and proved that the asymptotic mixing time is equal in two dimensions
subject to a reflection condition. A different boundary behavior exists for the D4
lattice in dimension 4, in which the open boundary can change the mixing time.
The methods include an analysis of the functions which are ‘harmonic modulo 1’
on a tiling graph and of the asymptotic behavior of the tiling’s Green’s function.
Jing-Jing Huang, University of Nevada, Reno
Title: The rational points close to a manifold
Abstract: We will mainly talk about problems related to counting rational points
in a thin neighborhood of a manifold. We have recently solved the case of hypersurfaces, and made some progress in the case of space curves. There are also significant
applications of our results to diophantine inequalities and metric diophantine approximation on manifolds.
Trevor Hyde, University of Michigan
Title: Cyclotomic factors of necklace polynomials
Abstract: Necklace polynomials enumerate aperiodic necklaces of colored beads.
They have a long history passing through number theory, geometry, representation
theory, and combinatorics. I will discuss some recent work which began with theobservation that necklace polynomials vanish at many roots of unity and eventually led to connections with multiplicative cyclotomic unit relations and the Euler
characteristics of the space of all multivariate irreducible polynomials.
Alex Iosevich, Univeristy of Rochester
Title: On certain analytic, combinatorial and number-theoretic aspects of frame
theoryAbstract: We are going to describe some recent results pertaining to the existence
and non-existence of exponential bases and frames and Gabor bases. We shallemphasize the plethora of techniques and ideas from several areas of mathematics
that go into the analysis of the underlying phenomena.
Mizan R. Khan, Eastern Connecticut State University
Title: A conjecture on visible points in lattice parallelograms
Abstract: Let a, n ∈ Z+ , with a < n and gcd(a, n) = 1. Let Pa,n denote the lattice
parallelogram spanned by (1, 0) and (a, n), that is,

Pa,n = {t1 (1, 0) + t2 (a, n) : 0 ≤ t1 , t2 ≤ 1} ,and let V (a, n) be the number of visible lattice points lying in the interior of Pa,n .
We will state some elementary (and straightforward) results for V (a, n). We conjecture that for a 6= 1, n − 1, V (a, n) satisfies the inequality
0.5n < V (a, n) < 0.75n.
The basic heuristic for the upper bound is that at least one-fourth of the lattice
points inside the parallelogram have both even x and y co-ordinates. However(other than extensive computer calculations) we do not have a heuristic for the
lower bound.
Joint work with Joydip Saha and Peng Zhao.
Sandor Kiss, Budapest University of Technology and Economics, Hungary
Title: A problem of Erdős about sets without pairwise coprime integers
Abstract: Let f (n, k) be the largest cardinality of the subsets of positive integers
not exceeding n which does not contain k +1 pairwise coprime integers. Let E(n, k)
be the set of positive integers up to n which have least one prime divisor among thefirst k primes. Erdős conjectured in 1962 that f (n, k) is equal to the cardinality ofE(n, k) for all large enough n. In my talk I am telling some of our recent results
about the conjecture.
Alex Kontorovich, Rutgers University, New Brunswick
Title: On working with Jean Bourgain
Abstract: We will give a personal/historical overview of some problems of interest
to Bourgain, and recent developments thereof.
Ben Krause, California Institute of Technology
Title: Discrete analogues in harmonic analysis: Directional maximal functions in
Z2
Abstract: Let V = {v1 , . . . , vN } be a collection of N vectors that live near a discrete
sphere. We consider discrete directional maximal functions on Z2 where the set of
directions lies in V , given byX

supv∈V,k≥C log N

f (x − v · n) · φk (n) , f : Z2 → C,n∈Z

where and φk (t) := 2−k φ(2−k t) for some bump function φ. In particular, we seek
to bound these operators on `2 (Z2 ) as efficiently as possible in terms of |V |.
Interestingly, the study of these operators leads one to consider an “arithmetic
version” of a Kakeya-type problem in the plane, which we approach using a combination of geometric and number-theoretic methods. Motivated by the Furstenberg
problem from geometric measure theory, we also consider a discrete directional
maximal operator along polynomial orbits,
sup
v∈V

X

f (x − v · P (n)) · φ(n) , P ∈ Z[−].

n∈Z

This is joint work with Laura Cladek.

Noah Kravitz, Yale University
Title: A stronger connection between the Erdős-Burgess and Davenport constants
Abstract: The Erdős-Burgess constant of a semigroup S is the smallest positive integer k such that any sequence over S of length k contains a nonempty subsequence
whose elements multiply to an idempotent element of S. In this talk, we focus on
the case where S is the multiplicative semigroup of Z/nZ. The following conjectureof Hao, Wang, and Zhang connects the Erdős-Burgess constant of Z/nZ (written
Ir (Z/nZ)) and the Davenport constant of (Z/nZ)× (written D((Z/nZ)× )):Ir (Z/nZ) = D((Z/nZ)× ) + Ω(n) − ω(n),where Ω(n) denotes the total number of primes in the prime factorization of n(with multiplicity), and ω(n) denotes the number of distinct primes dividing n.
Extending previous results for n squarefree and n a prime power, we prove thisconjecture when n is twice a prime power, n has exactly two prime divisors, or n isdouble the product of two odd prime powers. We also discuss the extension of ourtechniques to other rings.
Joint work with Ashwin Sah (MIT).
Paolo Leonetti, Graz University of Technology, Austria
Title: Small sets of integers
Abstract: We say that a function µ? : P(N+ ) → [0, 1] is an upper quasi-density if
it is subadditive, normalized (i.e., µ? (N+ ) = 1), and µ? (k · X + h) = k1 µ? (X) for
all X ⊆ N+ and k, h ∈ N+ , where k · X + h := {kx + h : x ∈ X}. Examples include
the upper Buck, upper Pólya, and upper analytic densities, as well as all the upper
α-densities (with α ≥ −1, subsuming the upper logarithmic and upper asymptotic
densities). A set X ⊆ N+ is said to be B-small if µ? (X) = 0 for every upper
quasi-density µ? . We establish that a set X is B-small if and only if it belongs to
the zero set of the upper Buck density. Then, we show that each of the following
sets are B-small: the primes; the perfect powers; the set of all n ∈ N+ such thatf (n) is prime, where f is a non-constant polynomial with integer coefficients; andthe set of integers which can be expressed as sum of two squares.
Huixi Li, University of Nevada, Reno
Title: On two lattice points problems about the parabola
Abstract: In this presentation I will talk about our recent results on the asymptotic
formulae with optimal error terms for the number of lattice points under and neara dilation of the standard parabola, the former improving upon a previous resultof Popov. These results can be regarded as achieving the square root cancellationin the context of the parabola, whereas its analogues are wide open conjectures
for the circle and the hyperbola. Our proofs utilize techniques in Fourier analysis,
quadratic Gauss sums and character sums.
Joint work with Jing-Jing Huang.

Neil Lyall, University of Georgia
Title: Hypercubes in sets of positive upper density
Abstract: We plan to discuss the following two results:
1. Any set A ⊆ R2d of positive upper density necessarily contains a d-dimensional
cube of sidelength λ, for all λ ≥ λ0 (A).
2. Any set A ⊆ Z5d of upper density δ > 0 necessarily contains a d-dimensional
cube of sidelength λ, for all λ ≥ λ0 (A) with the property that λ2 ∈ q(δ)Z. Results ofthis type also hold for more general orthotopes and generalize foundational resultsof Bourgain on non-degenerate simplices in subsets of Euclidean space of positive
upper density.
Joint work with Akos Magyar.
Akos Magyar, University of Georgia
Title: Spherical point configurations
Abstract: In the late 80’s Bourgain proved that sets of positive upper density of
Rn contain an isometric copy of all sufficiently large dilates of any non-degenerate
k-simplex. We show how to extend this result to spherical k-point configurations
spanning a k − 2-dimensional subspace, both in the model case of finite field geometries as well as in the Euclidean setting, using techniques from additive combinatorics.
Joint work with N. Lyall and H. Parshall.
Ariane Masuda, New York City Tech (CUNY)
Title: Functional graphs of Rédei functions
Abstract: Let Fq be a finite field of order q and f : Fq → Fq be a mapping. The
functional graph associated to f is a directed graph where the vertices are labelledby the elements of Fq and an edge connects a vertex a with a vertex b if and only
if b = f (a). In this talk we discuss the structure of the functional graph of a Rédeifunction Rn (x, a) over Fq where n ∈ N and a ∈ F∗q . We show conditions for suchgraphs to be isomorphic for a fixed q, and present several families of isomorphic
graphs.
Joint work with Juliane Capaverde and Virgı́nia Rodrigues.
Ryan W. Matzke, University of Minnesota - Twin Cities
Title: Maximum size (k, l)-sum-free sets in abelian groups
Abstract: A subset A of a finite abelian group G is called (k, l)-sum-free if the sumof k elements of A never equals the sum of l elements of A. Sets that satisfy this fork = 2 and l = 1 are often simply called sum-free sets. In 2005, Ben Green and Imre
Ruzsa found an explicit formula for the maximum size of a sum-free subset of any
finite abelian group. While no generalization of their result exists for finite abeliangroups, we shall present a recent result for finite cyclic groups for all k, l > 0, anddiscuss what is currently known.
Joint work with Béla Bajnok.

Steve Miller, Williams College
Title: Recent progress in sumsets, generalized Schreier sets, generalized Zeckendorf
decompositions, and prime and square-free Walks
Abstract: This is a report on work in progress from the 2018 SMALL REU anda Spring 2019 research class at Carnegie Mellon. Our first result extends work of
Lazarev, Miller and O’Bryant, who showed there was a divot in the probabilityof a sumset missing exactly k elements (with the set drawn uniformly from all
subsets of {0, ..., n}). We present some theorems and evidence for what happens
if instead each element is chosen with a fixed probability p. Our second result
concernces Generalized Schreier sets. A Schreier set is a set S of natural numberswith min S = |S|, and the Schreier family containing all Schreier sets is denoted
S1 . It has been known that for all n that the sequence of |{S ∈ S1 | max S = n}|is the Fibonacci sequence. By defining an α-Schreier set to be a set S of natural
numbers with | min S/a| = |S|, we generalize the result and prove that the sequenceof {S|maxS = n, | min S/a| = |S|} is of a linear recurrence relation of a higher order,
and also consider generalizations with additional constraints. If time permits wewill mention some recent results in Generalized Zeckendorf Decompositions andPrime and Square-Free walks (if not we will pose questions on these in the problem
session).
Joint work with Tudor Popescu (CMU) and Zimu Xiang (CMU)
Mariusz Mirek, Rutgers University - New Brunswick
Title: Dimension free estimates for the discrete Hardy–Littlewood maximal functions
Abstract: The aim of this talk is to present recent developments in dimension-freeestimates in discrete harmonic analysis. This topic has a very natural connection
with counting of lattice points in the Euclidean balls and with the Waring problem
for the squares. In particular, we show that the discrete Hardy–Littlewood maximal
functions associated with the Euclidean balls in Zd with dyadic radii have bounds
independent of the dimension on `p (Zd ) for every p ∈ [2, ∞].
Joint work with J. Bourgain, E.M. Stein, and B. Wróbel.
Akshat Mudgal, University of Bristol
Title: Sums of linear transformations in higher dimensions
Abstract : Given a finite subset A of integers and co-prime natural numbers q
and s, we consider the set q.A + s.A, that is, the sum of dilates of A. In recent
years, finding suitable lower bounds for the cardinality of such sets in terms of|A|, q, and s has seen considerable activity. In 2014, Balog and Shakan found sharp
estimates for the same, that were tight in both the main term as well as the error
term. Subsequently, they considered this problem in higher dimensional integer
lattices. In this talk, we present a short survey of these results including our own
improvement in the higher dimensional setting.

Rishi Nath, York College (CUNY)
Title: Simultaneous core partitions with nontrivial common divisor
Abstract: The last decade has a tremendous amount of activity around simultaneous (s, t)-core partitions when s and t have no common divisor. Here we relax thatcondition and explore new bijections and congruences for (s, t)-core and (s̄, t̄)-core
partitions when gcd(s, t) > 1.
Joint work with J.B. Gramain and J. A. Sellers.
Mel Nathanson, Lehman College (CUNY)
Title: Matrix scaling and diophantine problems
Abstract: The sequence of matrices obtained from a positive n × n matrix A by alternate row and column scaling will converge to a doubly stochastic matrix, called
the Sinkhorn limit of A. This leads to several problems in diophantine approximation. Moreover, computer experiments suggest that other natural variations ofalternate row and column scaling also converge to doubly stochastic matrices thatare different from the Sinkhorn limit. The meaning of these limits is unclear.
Timothy Newlin, United States Military Academy, West Point , NY
Title: Twitter response to Munich July 2016 attack: Network analysis of influence
Abstract: Social media platforms in cyberspace provide communication channels
for individuals, businesses, as well as state and non-state actors (i.e. individuals
and groups) to conduct messaging campaigns. This research analyzes the spheresof influence that arose around the keyword #Munich on Twitter following an active
shooter event at a Munich shopping mall in July 2016. Tweets utilizing #Munich
were captured beginning one hour after the shooting was reported and data collection ended approximately one month later. We constructed a cumulative networkand several daily networks from this data, and analyzed community evolution using
the Louvain algorithm. The analysis of how such communities change over time
has many applications to both encourage and discourage the effectiveness of an
information messaging campaign. The largest communities observed in the early
portions of the data disappear from the #Munich conversation within seven days.
The politically charged nature of many of these communities suggests their activity
migrated to other hashtags (i.e. conversation topics). Future analysis of Twitter
activity might focus on tracking communities across topics and time. Finally, wesuggest techniques on methods for constructing synthetic networks to further understand how these communities of influence adapt over time.
Joint work with Ivan Bermudez, Jan-Daniel Cleven, Ralucca Gera,Erik Kiser.
Kevin O’Bryant, College of Staten Island (CUNY)
Title: Explicit bounds on the number of zeros of Dirichlet L-functions
Abstract: We will present new explicit results on the number of zeros of Dirichlet
L-series with height at most T .

Andrew Odesky, University of Michigan
Title: Characterizing polynomial sequences in p-adic interpolation
Abstract: Consider a sequence s : N → k valued in a number field k. Given a place
ν of k lying over a prime p, one may ask under what circumstances the sequences is interpolated by a nice function f : Zp → kν , where kν is the completion of
k for the place ν. For instance, if s is given by the values of some polynomial,then it may be interpolated by ν-adic analytic functions for any place ν of k. The
converse to this example presents an interesting problem in p-adic analysis: How
to characterize the polynomial sequences among all sequences in terms of natural
p-adic conditions?
An elegant theorem of R.R. Hall from 1971 gives one satisfactory answer: If s
preserves congruences and its generating function has sufficiently large Archimedean
radius of convergence, then it is polynomial. Work of D.L. Hilliker and E.G. Straus
gives another answer when s may be interpolated by analytic functions. A theoremof B. Dwork from his famous paper on the rationality of the zeta function of analgebraic variety gives another converse result.
In this talk I will present new converse results of a general type which subsume
the results mentioned above.
Jun Seok Oh, University of Graz
Title: An inverse zero-sum problem for some non-abelian groups
Abstract: Let G be a finite group. By a sequence over G, we mean a finite unorderedsequence of terms from G, with repetition allowed. We say that it is a product-one
sequence if its terms can be ordered such that their product equals the identity
element of G. The large Davenport constant D(G) is the maximal length of aminimal product-one sequence, that is a product-one sequence which cannot be
factored into two non-trivial product-one subsequences.
The study of the Davenport constant of finite abelian groups is a central topic
in zero-sum theory since the 1960s. Both the direct problem, asking for the precise
value of the Davenport constant in terms of group invariants, as well as the associated inverse problem, asking for the structure of extremal sequences, have received
wide attention in the literature. Even in the abelian case, little is known beyond
groups of rank at most two.
In recent years, Geroldinger and Grynkiewicz derived the precise value the large
Davenport constant for groups having a cyclic index 2 subgroup, and dihedral
and dicyclic groups belong to this class of groups. In this talk, we provide explicit
characterizations of all minimal product-one sequences of length D(G) over dihedral
and dicyclic groups.
Joint work with Qinghai Zhong.
Yumeng Ou, Baruch College (CUNY)
Title: Applications of decoupling in distance set problems
Abstract: The Falconer distance set conjecture says that if a compact set in Rd
has Hausdorff dimension greater than d2 , then its distance set must have positive
Lebesgue measure. The conjecture remains open in all dimensions d ≥ 2. There
have been many new developments recently achieved towards it as well as somerelated distance set problems, and the celebrated Bourgain-Demeter decoupling
inequality played an important role. We will talk about some of these connections.
Nikita Pereverzin, United States Military Academy, West Point , NY
Title: The God’s Number for the n x n Rubik’s Slide
Abstract: God’s Number represents the maximum of the minimum numbers oftransformations from any initial configuration to any possible final configurationof a n x n Rubik’s Slide, if the game is played with no error. The different moves(shifting and rotating) form a group of permutations. Jones et al. leveraged thisgroup structure to develop different approaches varying in complexity to derive
the God’s Number. Jones et al. proved that the God’s Number is the diameterof the resulting Cayley Graph and also found a way to derive the God’s Number
from coset decompositions. Using these decompositions, Jones et al. was able tocalculate the God’s Number for smaller sized Rubik’s Slides.
The objective of this project is to advance the work of Jones et al. to find theGod’s Number for the n x n Rubik’s Slide. To this end, we built a scala programthat generates the adjacency matrix for any Rubik’s Slide and then calculates the
spectral radius of the matrix. Additionally, we developed a machine learning algorithm to simulate the Rubik’s Slide game for n = 2 and n = 3 and statisticallyestimate the God’s Number.
Cosmin Pohoata, California Institute of Technology
Title: Sets without 4APs but with many 3APs
Abstract: It is a classical theorem of Roth that every dense subset of {1, . . . , N }contains a nontrivial three-term arithmetic progression. Quantitatively, results ofSanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least
1/(log N )1− already have this property. In this talk, we will discuss the existenceof sets of N integers that, unlike {1, . . . , N }, do not contain nontrivial four-termarithmetic progressions, but still have the property that all subsets of relative density at least 1/(log N )1− must contain a three-term arithmetic progression. We
shall also discuss the analogous story over Fnq , where there exist four-term progression free sets A such that, for every subset A0 ⊂ A with |A0 | ≥ |A|1−c , the set A0
contains a three term arithmetic progression.
Joint work with Oliver Roche-Newton.
Yuneid Puig de Dios, University of California, Riverside
Title: Křı́ž’s theorem via dynamics of linear operators
Abstract: The existence of a set A ⊆ Z of positive upper Banach density such thatA − A := {m − n : m, n ∈ A, m > n} does not contain a set of the form E − E with
E piecewise syndetic is in essence the content of a popular result due to Křı́ž in1987, in which he used a graph-theoretical approach. More recently, other proofs ofthis result have been given using combinatorial number theory. Our goal here is toshow that a stronger result than the one given by Křı́ž can be obtained, and that
this can be done via operator theory, namely using dynamics of linear operators on
Banach spaces.

Alex Rice, Millsaps College
Title: Deligne polynomials in difference sets
Abstract: We will discuss strong upper bounds on the density of sets of integers
lacking differences of the form h(m, n), where h ∈ Z[x, y] satisfies appropriatesmoothness conditions inspired by finite field exponential sum estimates of Deligne.
Joint work with John Doyle.
Misha Rudnev, University of Bristol
Title: On growth in groups SL2 (Fp ) and Aff(Fp )Abstract: The talk will focus on the geometric basics that enable one to provereasonable quantitative estimates for the rate of growth, by group multiplication,in the above groups. The first qualitative results of this kind are due to Helfgott.
They are a fountainhead of today’s rich theory of finite approximate groups and its
applications, where several papers of Bourgain and collaborators played a pivotal
role. Apart from attempting to elucidate the structure of the proofs, the talk willaddress the relationship between growth in non-commutative groups and the sumproduct phenomenon.
Alisa Sedunova, MPIM Bonn
Title: Intersections of binary quadratic forms in primes and the paucity phenomenon
Abstract: Let r(n) be the function that counts the number of ways to represent anatural number n as a sum of two positive squares. The number of solutions to
a2 + b2 = c2 + d2 ≤ x, hence, the second moment of r(n), in integers is a well-known
result, while if one restricts all the variables to primes Erdős showed that only the
diagonal solutions, namely, the ones with a = c, b = d contribute to the main term,hence there is a pacuity of the off-diagonal solutions. In this talk we discuss thework in progress about second moments of r(n), when some of the variables are restricted to primes and discuss the existence of paucity for the off-diagonal solutions
in such problems. The methods are largely based on Hooley technique to tackle (onGRH) the Hardy-Littlewood problem about the representations N = p + a2 + b2 ,where p is a prime and a, b are integers, some related works of Plaksin (based on
the unconditional resolution of the Hardy-Littlewood problem by Linnik) and more
recent results of S. Daniel.
Steven Senger, Missouri State University
Title: Chains of points determined by distances
Abstract: We consider the following generalization of the celebrated unit distanceproblem: Given any large finite set of n points in the plane, and k distances,d1 , d2 ,. . . , dk , how many (k + 1)-tuples of points can there be where subsequent pairsof points are separated by the indicated distances. Such (k + 1)-tuples are called
chains. We present new upper and lower bounds for the maximum possible numberof chains. We also present related results in other settings: three dimensions, finite
fields, finite rings, and an analogous question about dot products.

Adam Sheffer, Baruch College (CUNY)
Title: Lower bounds for incidences with hypersurfaces
Abstract: Geometric incidences are a family of combinatorial problems that havea large variety of applications. Bourgain often relied on incidences, using them toobtain results in Number Theory (e.g., energy of Gaussian numbers), TheoreticalComputer Science (2-source extractors), and Harmonic Analysis (restriction problems). Bourgain and Demeter used incidences to study a discrete Fourier restriction
to the four- and five-dimensional sphere. We will show how reversing the directionof their proof leads to new lower bounds for a family of incidence problems. This
leads to one of few cases where an incidence problem is solved up to sub-polynomial
factors.
Ilya Shkredov, Steklov Mathematical Institute, Moscow, Russia
Title: SL2 –actions, modular hyperbolas, and Kloosterman sums
Abstract: We are going to talk about the connection between growth in SL2 (F)for various fields F and a series of problems of Number Theory and Additive Combinatorics such as Zaremba’s conjecture form the theory of continued fractions,distribution of points onto modular hyperbolas, and estimates of bilinear forms of
Kloosterman sums.
Satyanand Singh, New York City Tech (CUNY)
Title: Generating terms of Nathanson’s lambda sequences
Abstract: We consider the set∞[{εj (n) · nj : εj (n) ∈ {0, ±1, ±2, ..., ±bn/2c}}.
An =
j=0

S

Let SA = a∈A Aa , where A ⊆ N. We denote by λA (h) the smallest positive
integer that can be represented as a sum of h, and no less than h elements in SA .
Nathanson studied the properties of the λA (h) sequence and posed the problem offinding the values of λA (h), which are very important in geometric group theory
and additive number theory. When A = {2, j}, we represent λA (h) by λ2,j (h). For
fixed h ∈ {1, 2, 3}, the values of λ2,j (h) are known as j runs over the odd integers
bigger than 1. In this presentation, we extend this result further by illustrating how
to generate λ2,j (4) as j runs over the odd integers bigger than 1. In particular we
will show that λ2,3 (4) = 150 and λ2,5 (4) = 83. Our technique involve establishingthe insolubility of certain exponential diophantine equations.
Jozsef Solymosi, University of British Columbia, Canada
Title: Problems and results on the sum-product problem of Erdős and Szemerédi
Abstract: I will talk about the sum-product problem.

Jonathan Sondow, New York
Title: On Carmichael and polygonal numbers, Bernoulli polynomials, and sums ofbase-p digits
Abstract: We give a new characterization of the set C of Carmichael numbers in
the context of p-adic theory, independently of the classical results of Korselt andCarmichael. The characterization originates from a surprising link to the denominators of the Bernoulli polynomials via the sum-of-base-p-digits function. Moreprecisely, we show that such a denominator obeys a triple-product identity, where
one factor is connected with a p-adically defined subset S of the squarefree integers
that contains C. This leads to the definition of a new subset C 0 of C, called theprimary Carmichael numbers. Subsequently, we establish that every Carmichaelnumber equals an explicitly determined polygonal number of a special type. Finally, the set S is covered by modular subsets Sd (for d ≥ 1) that are related to the
Knoedel numbers, where C = S1 is a special case.
Joint work with Bernd C. Kellner. See arxiv.org/abs/1902.10672 for our preprint.
Wenbo Sun, Ohio State University
Title: Sarnak’s Conjecture for nilsequences on arbitrary number fields, and applications
Abstract: Sarnak’s conjecture is a hot topic in number theory and ergodic theory
in recent years. The conjecture says that the Möbius function is orthogonal to any
sequence coming from a dynamical system of low complexity. Nowadays, many
special cases of the Sarnak’s conjecture have been proven and many applications
have been found. In this lecture, we shall talk about generalizations of Sarnak’s
conjecture to the ring of integers of an arbitrary number field, and present new
results in this direction. We shall also talk about its application in combinatorics.
For example, we show that for any finite coloring of the Gaussian integers Z[i],there exist x and y distinct and non-zero of the same color such that x2 − y 2 = n2
for some Gaussian integer n.
Van Vu, Yale University
Title: On square sum-free sets
Abstract: How many numbers can one choose from {1, 2, ...n} so that no subset
sum is a square? This old problem of Erdős has a fascinating history, linking manyleading researchers of the field of additive combinatorics (some in a way you would
not expect).
Aled Walker, Trinity College, Cambridge
Title: Linear inequalities in primes
Abstract: For over 80 years it has been known how to use Fourier analysis to prove
an asymptotic formula for the number of three-term arithmetic progressions ofprimes less than N . These methods also apply to m simultaneous linear equationsin primes, provided that the number of prime variables is at least 2m + 1. Much
more recently, by the revolutionary work of Green-Tao-Ziegler, it was shown how
to count the number of solutions to such systems provided that there were at leastm + 2 prime variables. In this talk we will discuss the related setting of linearinequalities in primes. By adapting the methods of Green and Tao, together withsome additional ideas, we will prove an asymptotic formula for the number of prime
solutions to systems of m simultaneous inequalities with at least m + 2 variables.
This greatly improves upon the existing analytic methods, which, by analogy withlinear equations, require at least 2m + 1 variables.
Hong Wang, MIT
Title: Incidences between points and tubes
Abstract: We prove analogues of Szemerédi-Trotter theorem and other incidence
theorems for δ–approximate incidences, assuming that the δ–tubes are well-spaced
in a strong sense.
Joint work with Larry Guth and Noam Solomon.