CANT 2016 Abstracts
New York Number Theory Seminar
Fourteenth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 24–27, 2016

Sarfraz Ahmad, Comsats Institute of Information Technology, Lahore, Pakistan
Title: On the chain blockers of the poset C5 × Cb
Abstract: Let P = Ca × Cb be a poset where Ci is the i-element chain 1 < · · · < i
for i ≥ 1. A chain blocker for P is defined as a subset A ⊆ P such that A is inclusionwise minimal with the property that every maximal chain in P contains at least
one element of A. We discuss new combinatorialinterpretation of the convoluted

2n−k
k
and
Catalannumbers are introduced in
Catalan numbers C(n, k) := 2n−knterm of the chain blockers.
We are interested in chains blockers as they appear for a poset P as the minimal
nonfaces of simplicial complex ∆∗P which is Alexander dual to the simplicial complex∆P whose minimal non-faces are the maximal chains in P . Recall that a simplicial
complex ∆ over a ground set Ω is a subset ∆ ⊆ 2Ω such that A ⊆ B ∈ ∆ impliesA ∈ ∆. In this talk we will give a complete description of the chain blockers of
C5 × Cb for b ≥ 1.

Paul Baginski, Fairfield University
Title: Arithmetic progressions, nonunique factorization, and additive combinatorics
in the group of units mod n
Abstract: For integers 0 < a ≤ b, the arithmetic progression Ma,b := a+bN is closed
under multiplication if and only if a2 ≡ a mod b. Any such multiplicatively closedarithmetic progression is called an arithmetic congruence monoid (ACM). Thoughthese Ma,b are multiplicative submonoids of N, their factorization properties differ
greatly from the unique factorization one enjoys in N.
In this talk we will explore the known factorization properties of these monoids.
When a = 1, these monoids are Krull and behave similarly to algebraic number
rings, in that they have a class group which controls all the factorization. Combinatorially, factorization properties correspond to zero-sum sequences in the group.
However, when a > 1, these monoids are not Krull and thus do not have a classgroup which fully captures the factorization behavior. Nonetheless, an ACM canbe associated to a finite abelian group, whose additive combinatorics relate to thefactorization properties of the ACM. We will pay particular attention to the factorization property of elasticity and its connection to sequences in the group which
attain certain sums while avoiding others.
1Gautami Bhowmik, Universit of Lille, France
Title: A combinatorial problem from a mediaeval Sanskrit text
Abstract: We will explain the construction and enumeration of pan-diagonal magic
squares as they appear in Ganitakaumudi (1356).

Pierre Bienvenu, University of Bristol, UK
Title: Higher dimensional Siegel-Walfisz theorem
Abstract: The Green-Tao theorem provides asymptotics for the number of prime
tuples of the form (ψ1 (n), . . . , ψt (n)) when n ranges among the integer vectors of aconvex body K ⊂ [−N, N ]d and Ψ = (ψ1 , . . . , ψt ) is a system of affine linear forms
whose linear coefficients remain bounded (in terms of N ). In the t = 1 case, theSiegel-Walfisz theorem shows that the asymptotic still holds when the coefficientsvary like a power of log N . We prove a higher dimensional (i.e. t > 1) version ofthis fact. To do so, we basically rely on the work on Green and Tao, but inject
new ideas to overcome the substantial use they make of the boundedness of thecoefficients.

Arnab Bose, University of Lethbridge, Canada
Title: Investigations on some exponential congruences
Abstract: Around 1981, Selfridge asked for what positive integers a and b witha > b, does 2a − 2b divide na − nb for all n ∈ N. The problem was independently
solved by various people in different contexts, notably C. Pomerance (1977), Sun
Qi and Zhang Ming Zhi (1985). In this talk, we study their ideas and prove a
generalization of the problem, in the elementary number theoretic sense and also in
algebraic number fields. Further, we develop ideas to give a conditional resolution
and generalizations to another problem by H. Ruderman which is closely related to
Selfridge’s problem. (Joint work with Amir Akbary)

Kamil Bulinski, University of Sydney, Australia
Title: Twisted Multiple Recurrence and patterns in large subsets of Zd
Abstract: Recurrence is a fundamental notion in Ergodic Theory. It has a strong
relationship with combinatorial number theory, as was first shown by Furstenberg
when he proved a multiple recurrence result and deduced from it Szemerédi’s theorem on arithmetic progressions. After a brief review of the relationship between
recurrence and the existence of configurations in dense sets, I will present some new“twisted” multiple recurrence results obtained with M.Björklund (Chalmers) andtheir applications to finding “twisted” configurations in positive density subsets of
Zd . This has many corollaries on the structure of images of positive density subsets
of Zd under quadratic forms, as well as other homogeneous polynomials with a large
group of symmetries.

Sam Cole, University of Illinois at Chicago
Title: Planted partitions in random graphs
Abstract: In the planted partition problem, n = ks vertices of a random graphare partitioned into k unknown “clusters,” each of size s. Edges between vertices
in the same cluster and different clusters are included with constant probability p
and q, respectively (where 0 ≤ q < p ≤ 1). The goal is to recover the unknown
clusters from the randomly generated graph. I will give a brief survey of results forthis problem and present a simple spectral algorithm that, with√high probability,
recovers the partition as long as the cluster sizes are at least Ω( n).

Colin Defant, University of Florida
Title: The unitary Cayley graph of Z/nZ
Abstract: If R is a commutative ring with 1, then the unitary Cayley graph of R,
denoted GR , is the graph with vertex set V (GR ) = R and edge set
E(GR ) = {{x, y} : x − y ∈ R× }.
In particular, if x, y ∈ Z/nZ, then x and y are adjacent in GZ/nZ if and only if
x − y is relatively prime to n. In this talk, we give a simple formula for the numberof cliques of order m in GZ/nZ for any positive integers m and n. This formula
involves a class of arithmetic functions, known as Schemmel totient functions, thatgeneralize the Euler totient function. We also discuss attempts to determine thedomination numbers of the graphs Z/nZ. This problem is related to the Jacobsthal
function j.

Mohamed El Bachraoui, United Arab Emirates University 
1−q b nTitle: On the polynomiality of 1−qa m
 
1−q b nAbstract: In this talk we will give sufficient conditions for 1−q
to be a polya mnomial with nonnegative integer coefficients. This unifies a variety of special cases
already known in literature.

George Grossman, Central Michigan University
Title: Combinatorial identities with Fibonacci and Lucas numbers
Abstract: In this talk we explore aspects of combinatorial identities. In particular,
it is shown that each Fibonacci number, up to sign change, can be represented
countably many distinct ways as sum of binomial coefficients.

Sandie Han, New York City Tech (CUNY)
Title: Characterization of matrix ∈ GL2 (N0 ) as product of certain pairs of 2 × 2
matrices
Abstract: This is continuation of the study on the characterization of the posiaz + b
tive linear fractional transformation (PLFT) of the formin the generalizedcz + dCalkin-Wilf tree, wherea, b, c, d are non-negative integers and ad − bc 6= 0.
 The
a b
1 v
associated matrix
can be represented as product of A(v) =
, and
c d
0 1


1 0
B(u) =, for some integers u, v ≥ 1.
u 1
Joint work with Masuda, Singh, and Thiel.

Brian Hopkins, St. Peter’s University
Title: Color restricted n-color compositions
Abstract: Following work of Agarwal and Andrews on the analogous partitions, in2000 Agarwal defined n-color compositions, where a part k can have one of k different colors. The dozen or so subsequent papers on this topic have primarily focusedon the n-color compositions that arise when only certain parts are allowed. Here,
we focus instead on restricting colors. The resulting sets of n-color compositions
provide combinatorial interpretations for many known integer sequences. We havedeveloped recurrence relations for three very general families of color restricted ncolor compositions and, for several particular cases, direct formulas and bijections
using the idea of spotted compositions introduced by the author in 2013.
Joint work with Hua Wang of Georgia Southern University.

Alex Iosevich, University of Rochester
Title: On analytic, combinatorial and number theoretic aspects of the distancegraph
Abstract: Let R be an abelian ring, finite or infinite, and let E ⊂ Rd , a ddimensional module over R. Given a non-zero t ∈ R, define the distance graph
Gt (E) by taking elements of E to be the vertices, and connecting two vertices
22x, y ∈ Rd by an edge if (x1 − y1 ) + · · · + (xd − yd ) = t. We shall examine somebasic properties of this graph when R = R, Z and Zp . A variety of diverse andinteresting mathematics quickly comes to the surface in response to even the mostbasic questions pertaining to this object.

William J. Keith, Michigan Technological University
Title: Partitions simultaneously regular, distinct, and/or flat
Abstract: Partitions with parts not divisible by m, appearing less than m times, orwith first differences less than m (and first part less than m) form three equinumerous classes, mapped by Glaisher’s map and conjugation. These classes have beenpart of the study of partitions since Euler’s initial work in the area. Yet few if any
results have been found for partitions fixed by these maps, which satisfy two or
three of these conditions, except for the classical result that partitions into distinct
odd parts are in bijection with the self-conjugate partitions. In this talk we explore
some of their properties and produce some results and questions.

Mizan Khan, Eastern Connecticut State University
Title: On White’s characterization of empty lattice tetrahedra
Abstract: In 1963 White gave an elegant characterization of lattice tetrahedra that
did not contain any lattice points other than their vertices. We will discuss a proofof White’s theorem due to Karen Rogers.

S. V. Konyagin, Steklov Mathematical Institute of the Russian Academy of Sciences
Title: Number of nontrivial solutions of an equation with reciprocals
Abstract: I will discuss our joint results with M. A. Korolev on an upper estimate
for the number of nontrivial solutions to the equation
11
1
1+ ··· +=
+ ··· +x1
xr
y1
yr
where x1 , . . . , xr , y1 , . . . , yr are positive integers not exceeding N . A solution is said
to be nontrivial if {y1 , . . . , yr } is not a permutation of {x1 , . . . , xr }.

Ben Krause, University of British Columbia
Title: Discrete analogues in harmonic analysis: A monomial Carlson theorem
Abstract: We give sufficient conditions on Λ under which the discrete maximal
functions below are bounded on `2 (Z).
d

Cd,Λ f (n) := sup
λ∈Λ

X
m6=0

f (n − m)e2πiλmm, d ≥ 2.

The set Λ can be certain kinds of Cantor sets, for instance. The integral versionof this result holds with no restriction on Λ, and is due to Eli Stein and Stephen
Wainger.
Joint work with Michael Lacey.

Urban Larsson, Dalhousie University, Halifax, Canada
Title: Sumsets and impartial games
Abstract: This talk concerns the star-operator of impartial vector subtraction
games. We show that reflexive (limit) games satisfy certain properties of sumsets, depending on the convention being either normal- or misere play.
Joint work with Silvia Heubach and Matthieu Dufour.
Jiange Li, University of Delaware
Title: An entropy analogue of the more sums than differences problem
Abstract: We study the comparison of Shannon entropies of sum and difference of
i.i.d. random variables. We show that they can differ by an real number additively,
but not too much multiplicatively. The study is related to the more sums thandifferences problem in additive combinatorics.
Ray Li, Carnegie-Mellon, Steven J. Miller, Williams College, Zhao Pan, CarnegieMellon, and Huanzhong Xu, Carnegie-Mellon
Title: Convergence rates in generalized Zeckendorf decomposition problems
Abstract: We report on two projects on generalized Zeckendorf decompositions.
The first concerns rates of convergence. Zeckendorf proved any integer can be decomposed uniquely as a sum of non-adjacent Fibonacci numbers, Fn . For Fibonacci
numbers, the fraction of gaps that are size k, Pk (k ≥ 2), over all m ∈ [Fn , Fn+1 )approaches 1/φk as n → ∞. Recently Bower et al. provided an explicit computation of Pk for all positive linear recurrence sequences {Gn }n∈N from the recurrence
relation. We extend these results and prove that, if Pk (n) denotes the fraction
k+O(1)| = n+O(1). Furtherof gaps that are size k for m ∈ [Gn , Gn+1 ), then |1 − PkP(n)kmore, we explicitly compute the O(1) terms in the case of the Fibonacci numbers,
and discuss its determination in many cases. We also report on some interesting
numerical observations on approximating these gap probabilities, and discuss thedetermination of a transition window for the observed behavior.
The second project concerns positive linear recurrence sequence (PLRS), which
are of the form Hn+1 = c1 Hn +· · ·+cL Hn+1−L with each ci ≥ 0 and c1 cL > 0, withappropriately chosen initial conditions. There is a notion of a legal decomposition
generalizing the non-adjacency condition from the Fibonacci numbers (roughly,
given a sum of terms in the sequence we cannot use the recurrence relation to reduceit) such that every positive integer has a unique legal decomposition of terms in thesequence. Previous work proved not only that a decomposition exists, but that the
number of summands Kn (m) in legal decompositions of m ∈ [Hn , Hn+1 ) converges
to a Gaussian. Using partial fractions and generating functions it is easy to show
the mean and variance grow linearly in n: an+b and cn+d; the difficulty is proving
a and c are positive. Currently the only way to do this requires delicate analysisof polynomials related to the generating functions and characteristic polynomials,
and is algebraically cumbersome. We introduce new, elementary techniques thatbypass these issues. The key insight is to use induction and bootstrap bounds
through conditional probability expansions to show the variance is unbounded, andhence c > 0 (the mean is handled easily through a simple counting argument).
These arguments can be generalized to other sequences that have a notion of legal
decomposition and unique decomposition.

Neil Lyall, University of Georgia
Title: Some new results in geometric Ramsey theory
I plan to give a quick introduction to geometric (density) Ramsey theory, introduce
a new perspective on the known results in the field (which lead to simplified proofs),
and ultimately discuss new results, in particular, the fact that (geometric) squares
are density Ramsey.

Akos Magyar, University of Georgia
Title: A Roth type theorem for dense subsets of Rd
Abstract: It is shown that dense subsets of Rd contain 3-term progressions of all
sufficiently large gaps, when the gap size is measured in the `p -metric for p > 1 andp not equal to 2. This is known to be false in the Euclidean `2 -metric as well as inthe `1 -metric, and one of our goal is to understand this phenomenon.
Joint work with B. Cook and M. Pramanik.

Ariane Masuda, New York City Tech (CUNY)
Title: Weierstrass semigroups on Kummer extensions
Abstract: In this talk we will show how one can compute the Weierstrass semigroupat one totally ramified place and at two certain totally ramified places for a Kummer
extension. We will also discuss some examples of one- and two-point Goppa codeswith good parameters derived from our results.
Joint work with Luciane Quoos and Alonso Sepúlveda.

Nathan McNew, Towson University
Title: Counting Integers divisible by a large shifted prime
Abstract: In 1980 Erdos and Wagstaff showed that most positive integers are notdivisible by any “large” shifted primes. We improve upon this result by obtaining
precise estimates for the count of integers up to x divisible a shifted prime p − 1 > yin essentially the full range of x and y. In particular, we show that as x and y tend
to infinity this count is at most x/(log y)b+o(1) , where b = 1 − (1 + log log 2)/ log 2
is the Erdős-Ford-Tenenbaum constant, and that this bound is optimal for a large
range of y.

Steven J. Miller, Williams College, Victor Xu, Carnegie-Mellon, and Xiaorong
Zhang, Carnegie-Mellon
Title: Existence of MSTD subsets and divots in the missing sums distribution
Abstract: We report on two projects on More Sum Than Difference (MSTD) sets.
For the first, let S be a finite subset of Z, and define the sumset S + S = {s + s0 :s, s0 ∈ S} and the difference set S −S = {s−s0 : s, s0 ∈ S}; S is called an MSTD set
if |S + S| > |S − S|. We investigate the existence of MSTD sets in subsets of Z. We
first prove that the Fibonacci sequence contains no MSTD subsets. Then we derivea sufficient condition for an integer sequence to have only finitely many MSTD
subsets based on the growth rate of the sequence. Using a standard number theory
conjecture, we are able to show that the set of prime numbers contains infinitely
many MSTD sets.
For the second, we choose A uniformly at random from subsets of {0, . . . , n − 1},
let A + A = {x + y : x, y ∈ A}, and let mn (A) denote the number of missing
sums in A + A (i.e., 2n − 1 − |A + A|). Previous work showed that as n → ∞the distribution of mn (A) has a divot at 7: m(7) < m(6) < m(8) (where m(k)denotes the limiting probability); that is, a random subset A is more likely to haveits sumset A + A missing 6 or 8 elements than missing 7 elements. We investigate
what happens under different models of choosing A; each element of {0, . . . , n−1} is
independently chosen to be in A with probability p (our initial model is equivalent
to p = 1/2). We show that if p > .966 then as n → ∞ there is a divot at 1. Further
research is ongoing to determine the movement of the divot as the probability of
selecting elements changes, and if and when there are multiple divots for certain
probabilities p.

Amanda Montejano, UMDI-Facultad de Ciencias, Universidad Nacional Autónoma
de México, Querétaro, México
Title: Zero-sum k-blocks in {−1, +1} bounded sum sequences
Abstract: In this talk, we consider problems and results that go in the opposite
direction of the classical theorems in the discrepancy theory. The following statement gives a flavor of our approach. Let t, k and q be integers such that q ≥ 0,
0 ≤ t < k, and t ≡ k (mod 2), and take s ∈ [0, t + 1] as the unique integer satisfying(mod (t + 2)). Then, for any integers ≡ q + k−t−2
2
q−s
t
1
k2 +
k− +s
2(t + 2)t+2
2Pn
and any function f : [n] → {−1, 1} with |P i=1 f (i)| ≤ q, there is a block of knconsecutive terms (k-block) B ⊂ [n] with | x∈B f (x)| ≤ t. Moreover, this boundis sharp for all the parameters involved and a characterization of the extremal
sequences is given. For example, taking t = 0, k = 6 andPn q = 2, we obtain that,
if n ≥ 15 then any function f : [n] → {−1, 1} with | i=1 f (i)| ≤ 2 contains azero-sum 6-block, and this is not true for n = 14, where the only function (up to
multiplication by −1) without zero-sum 6-blocks can be represented in the followingway:− − + + + + − − + + + + − −,n≥where the +’s and −’s represent the values +1 or, respectively, −1 at the corresponding position. This is a joint work with Yair Caro and Adriana Hansberg.

Brendan Murphy, University of Rochester
Title: Half-way to Szemeredi-Trotter over finite fields
Abstract: Recently, Roche-Newton, Rudnev, and Shkredov proved strong sumproduct estimates over finite fields using Rudnev’s point-plane incidence bound.
We present a generalization of their method based on a bound for the number of“collisions of images of lines.” This bound allows us to prove that the point set
P = A × A determines at most O(|A|9/2 ) collinear triples, which is halfway between
the trivial bound of O(|A|5 ) and the sharp bound of O(|A|4 log |A|) provided by theSzemeredi-Trotter theorem, valid over R and C.
Joint work with Esen Aksoy-Yazici, Misha Rudnev, and Ilya Shkredov.

Rishi Nath, York College (CUNY)
Title: Sequences of simultaneous core partitions
Abstract: A simultaneous (s, t)-core partition is one in which neither s nor t appears
in the set of associated hook numbers. We discuss some new results on simultaneous(s, ms±1)-cores and (2k−1, 2k+1)-cores and discuss some enumerative coincidences
for which mappings have yet to be found.

Mel Nathanson, Lehman College (CUNY)
Title: Comparison estimates for linear forms
Abstract: Let M be an R-module. Consider the h-ary linear form Φ : M h → M
Phdefined by Φ(t1 , . . . , th ) = i=1 vi ti For every subset A of M , defineΦ(A) = {Φ(a1 , . . . , ah ) : (a1 , . . . , ah ) ∈ Ah }.
with nonzero coefficient sequence (v1 , . . . , vh ) ∈ RhP. For every nonempty subset Iof {1, 2, . . . , h}, define the subsequence sum sI = i∈I vi . Let S(Φ) = {sI : I ⊆{1, 2, . . . , h}, I 6= ∅} be the set of all nonempty subsequence sums of the sequenceof coefficients of Φ. Let R× be the group of units of the ring R.
Theorem. Let Υ and Φ be linear forms with coefficients in the ring R. If 0, 1 ∈S(Υ) and if S(Φ) ⊆ R× , then for every ε > 0 there exists a finite R-module M anda subset A of M such that
Υ(A) = M
and|Φ(A)| < ε|M |.
Ph
For example, the linear forms Υ(t1 , t2 ) = t1 − t2 and Φ(t1 , . . . , th ) = i=1 tisatisfy the Theorem with R = Z or R = Z/mZ, where p > h for every prime
divisor p of m.

Péter Pál Pach, Budapest University of Technology and Economics, Hungary
Title: Progression-free sets in Zn4
Abstract: We show that if√ the subset A ⊆ Zn4 is free of three-term arithmetic
progressions, then |A| < 2( n + 1)4γn , with an absolute constant γ ≈ 0.926. Thatis, progression-free sets in Zn4 are exponentially small.
Joint work with Ernie Croot and Vsevolod F. Lev.

Giorgis Petridis, University of Rochester
Title: Products of differences in finite fields
Abstract: We present recent progress on a sum-product in finite fields question.
The question is to consider a set A in the prime order finite field Fp and ask how
large must its cardinality be so that the set of products of differences of elementsof A(A − A)(A − A) = (a − b)(c − d) : a, b, c, dinA
is large?
Bennet, Hart, Iosevich, Pakianathan, and Rudnev showed that |A| > p2/3 implies(A − A)(A − A)contains half the elements of Fp . This result is typical of a long listof sum-product questions in Fp , where the condition |A| > p2/3 implies that a set
derived by a combination of sums and products of elements of A has at least p/2
elements.
The purpose of the talk is to explain how prove the existence of an absolute
constant c¿0 such that |A| > p2/3−c implies that (A − A)(A − A) has at least p/2
elements. To the best of our knowledge this is the first instance in the literature
where the p2/3 threshold is beaten for such a type of sum-product question.
One of the ingredients is a recent result of Aksoy-Yazici, Murphy, Rudnev, andShkredov, which allows one to have a not tiny c.

Sinai Robins, ICERM at Brown University, and University of Sao Paulo, Brazil
Title: Discrete volumes of polytopes, Fourier transforms of polytopes, and solid
angle sums
Abstract. A discretized version of the volume of a polytope may be given by
the number of integer points in P. There are infinitely many different families of
discretized volumes of polytopes, but a particularly elegant one is the “solid anglesum” of a polytope, defined as follows. Place a very small sphere, centered ateach lattice point of a given lattice L ⊂ Rd , and consider the proportion of thatsphere that intersects the polytope P, called a local solid angle. Now translate thissmall sphere to an arbitrary lattice point of L, and again consider the local solid
angle contribution, relative to P. If we sum all of these contributions, over all lattice
points of L, we get the ”solid angle sum” of P, a nice discrete measure of the volumeof P. It turns out that if we dilate P by an integer t (called the dilated polytope tP)
and assume that the vertices of P lie on the lattice L, then this solid angle sum is apolynomial in the positive integer parameter t, and this polynomial is traditionallycalled AP (t) and first studied in detail by I. G. Macdonald. The coefficients ofthis polynomial encode certain geometric and number-theoretic properties of thepolytope P, but they are still mysterious and not easy to compute in general.

Here we extend the theory of these solid angle sums to all positive real dilationst, for any real polytope (so that its vertices do not necessarily lie on the latticeL). One of our ingredients is a detailed description of the Fourier transform of thepolytope P. This transform method uses the formula of Stokes, which is a wayof integrating by parts in several variables. Another key tool for us is the Poissonsummation formula, applied to smoothings of the indicator function of the polytope
P. It turns out that the combinatorics of the face poset of P plays a central role inthe description of the Fourier transform of P, and in keeping track of the infinite
series that pop out of Poisson summation. We also obtain a closed form for thecodimension-1 coefficient of the solid angle polynomial AP (t), extending previously
known results about this codimension-1 coefficient. This is joint work with Ricardo
Diaz and Quang-Nhat Le.

Bradley Rodgers, University of Michigan
Title: Divisor sums in short intervals and counts of plane partitions
Abstract: In this talk we will discuss recent joint work with Jon Keating, Edva
Roditty-Gershon, and Zeev Rudnick regarding the distribution of sums of k-folddivisor functions over a random short interval. We will talk about new conjectures
for the variance of these sums, and discuss an analogous result in a function fieldsetting which motivates the conjectures and which has a surprising connection toplane partitions.

Ryan Ronan, CUNY Graduate Center
Title: An asymptotic for the growth of Markoff-Hurwitz tuples
Abstract: Consider the Markoff-Hurwitz equation
x21 + x22 + · · · + x2n = ax1 x2 · · · xn + k
for integer parameters n ≥ 3, a ≥ 1, k ≥ 0. In this talk, we establish an asymptotic
count for the number of integral solutions with max{x1 , x2 , . . . , xn } ≤ R. Whenn = 3, a = 1 (or, equivalently, a = 3), and k = 0 this equation is known simply
as the Markoff equation, for which the asymptotic count was studied in detail by
Zagier in 1982. The previous best result for n ≥ 4 is due to Baragar in 1998
who established an exponential rate of growth with exponent β > 0 which is not,in general, an integer. We use methods from symbolic dynamics to improve thisasymptotic count, and which yield a new interpretation of this exponent β in termsof the unique parameter for which there exists a certain conformal measure on
projective space.
Joint work with Alex Gamburd and Michael Magee.

Tom Sanders, Oxford University, UK
Title: Monochromatic sums and products
Abstract: We shall discuss work with Ben Green on finding monochromatic quadruples (x, y, x + y, xy) in finite colourings of Fp .

James Sellers, Pennsylvania State University
Title: Infinitely many congruences modulo 5 for 4-colored Frobenius partitions
Abstract: In his 1984 AMS Memoir, Andrews introduced the family of functionscφk (n), which denotes the number of generalized Frobenius partitions of n into
k colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several
Ramanujan-like congruences for cφ4 (n) relative to different moduli. In this paper,
which is joint work with Michael D. Hirschhorn of UNSW, we employ classicalresults in q-series, the well-known theta functions of Ramanujan, and elementary
generating function manipulations to prove a characterization of cφ4 (10n + 1) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied
by cφ4 . This work greatly extends the recent work of Xia on cφ4 modulo 5.

Steven Senger, Missouri State University
Title: A few sum-product type estimates
Abstract: We will look at sets of the type AA + 1 and (A − A)(A − A), where A isa subset of either the real numbers or a finite field.
Satyanand Singh, New York City Tech (CUNY)
Title: The odd behavior of Nathanson’s lambda sequences
Abstract: For A any finite set of positive integers greater than 1, and a ∈ A we define the set Aa = {εj (a) · aj : j = 0, 1, 2, ...}, where εj (a) ∈ { 0, ±1, ±2, ..., ±ba/2c}.
Nathanson studied the properties of the related λA (h) sequences. In thisS settingpositive integers are partitioned as sums of elements from the set SA = a∈A Aa .
Nathanson posed the problem to compute λA (h), where λA (h) is defined as thesmallest positive integer that can be represented as the sum of elements of SA
with length h, but that cannot be represented as a sum with length less than h.
In this presentation we will restrict our study to sets of the shape A = {2, n} andodd n > 1 and illustrate how to generate the values of λ2,n (h) for fixed h ∈ {1, 2, 3}.

Jonathan Sondow, New York
Title: Babbage’s (non)-primality tests
Abstract: In 1819 Charles Babbage published his only paper in number theory. I
will recall his non-primality and primality tests, involving congruences for certain
binomial coefficients. I will mention the known counterexamples–Wolstenholme
primes squared–to the converse of Babbage’s non-primality test, and prove a partial
converse. Finally, I will give a generalization of his primality test.

Yoni Stanchescu, Afeka College and the Open University of Israel
Title: Small sumsets in torsion-free groups
Abstract: This talk will review some results and problems concerning small sumsets
in torsion-free groups. We will discuss recent advancements in different classes ofabelian and non-abelian groups. We will describe the structure of the extremal sets
and we are going to prove some precise structure theorems for sets of lattice pointsof small doubling.
Johann Thiel, New York City Tech (CUNY)
Title: The growth of coefficients in certain PLFT (u, v)-Calkin-Wilf trees
Abstract: A positive linear fractional transformation (PLFT) is a function of theform f (z) = az+b
cz+d where a, b, c, and d are nonnegative integer coefficients withdeterminant ad − bc 6= 0. Nathanson defined a PLFT (u, v)-Calkin-Wilf tree, withu, v positive integers, as an infinite rooted binary tree where every vertex is labelled
by a PLFT using a simple set of rules. If a vertex is labelled by the PLFT f (z), thenthe left child of the vertex is labelled by Lu (f (z)) and the right child is labelled byz
and Rv (f (z)) = z + v. In this talk we study the size
Rv (f (z)) where Lu (z) = uz+1of the coefficients of PLFTs appearing in certain PLFT (u, v)-Calkin-Wilf trees.
Joint work with Sandie Han, Ariane M. Masuda, and Satyanand Singh.

Andrew Treglown, University of Birmingham, UK
Title: Maximal sum-free subsets in the integers
Abstract: Cameron and Erdős asked whether the number of maximal sum-free setsin {1, . . . , n} is much smaller than the number of sum-free sets. In the same paper
they gave a lower bound of 2bn/4c for the number of maximal sum-free sets. We
prove the following: For each 1 ≤ i ≤ 4, there is a constant Ci such that, given any
n ≡ i mod 4, the set {1, . . . , n} contains (Ci + o(1))2n/4 maximal sum-free sets.
Our proof makes use of container and removal lemmas of Green, a structural resultof Deshouillers, Freiman, Sós and Temkin and a recent bound on the number ofsubsets of integers with small sumset by Green and Morris.
Joint work with Jozsef Balogh, Hong Liu and Maryam Sharifzadeh.

Yuri Tschinkel, NYU
Title: Counting integral points on orbits of algebraic groups
Abstract: I will describe recent results and conjectures concerning the distributionof integral points of bounded height on orbits of algebraic groups.
Joint work with A. Chambert-Loir.
Van Vu, Yale University
Title: Sum-free sets in groups
Abstract: We discuss several questions concerning sum-free sets in groups, raisedby Erdős in his survey “Extremal problems in number theory” (Proceedings of theSymp. Pure Math. VIII AMS) published in 1965. Among other things, we givea characterization for large sets A in an abelian group G which do not contain asubset B of fixed size k such that the sum of any two different elements of B do notbelong to A (in other words, B is sum-free, or sum-avoiding, with respect to A).
Erdős, in the above mentioned survey, conjectured that if |A| is sufficiently large
compared to k, then A contains two elements that add up to zero. This is known
to be true for k ≤ 3. We give counterexamples for all k ≥ 4. On the other hand,using the new characterization result, we are able to prove a positive result in thecase when |G| is not divisible by small primes.
Joint work with Terence Tao