CANT 2018 Abstracts
Sixteenth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 22–25, 2018

Sukumar Das Adhikari, Harish-Chandra Research Institute, and Ramakrishna
Mission Vivekananda Educational and Research Institute, India
Title: Some early Ramsey-type theorems in combinatorial number theory and some
generalizations
Abstract: We start with some classical results in Ramsey Theory and report somerecent results on some questions related to monochromatic solutions of linear Diophantine equations. We shall also mention few challenging open questions in thisarea of Ramsey Theory.
Shabnam Akhtari, University of Oregon
Title: Representation of integers by binary forms
Abstract: Let F (x, y) be an irreducible binary form of degree at least 3 and withinteger coefficients. By a well-known result of Thue, the equation F (x, y) = m has
only finitely many solutions in integers x and y. I will discuss some quantitative
results on the number of solutions of such equations. I will also talk about myrecent joint work with Manjul Bhargava, where we show that many equations of
the shape F (x, y) = m have no solutions.
Paul Baginski, Fairfield University
Title: Nonunique factorization in the ring of integer-valued polynomials
Abstract: The ring of integer-valued polynomials Int(Z) is the set of polynomials
with rational coefficients which produce integer values for integer inputs. Specifically,
Int(Z) = {f (x) ∈ Q[x] | ∀n ∈ Z f (n) ∈ Z}.
Int(Z) constitutes an interesting example in algebra from many perspectives; forexample, it is a natural example of a non-Noetherian ring. It is also a ring withnonunique factorization. Every element has only finitely many factorizations, yet
the number of irreducibles involved grows without bound. Frisch recently demonstrated that in Int(Z), you can find an element f (x) that has any factorization
lengths you desire and you can even prescribe the number of factorizations of eachlength. The polynomials constructed in this way have high degree. We give a
graded analysis, determining all the possible elasticities and catenary degrees fora polynomial as a function of the degree of the polynomial. Joint work with Greg
Knapp, Jad Salem, and Gabrielle Scullard.
1

Gautami Bhowmik, University of Lille, France
Title: Asymptotics of Goldbach functions and zeros of L–functions
Abstract: We show that good asymptotics of the classical Goldbach generating
function is equivalent to the Riemann Hypothesis. For the case of primes in arithmetic progression, we obtain information on zeros of Dirichlet L functions.
Margaret Bilu, New York University
Title: Motivic Euler products
The Grothendieck group of varieties over a field k is the quotient of the free abeliangroup of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties
over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will explain how some power series withcoefficients in the Grothendieck ring can be endowed with an Euler product decomposition analogous to classical Euler products in number theory, and discussgeometric applications of this notion.
Scott Chapman, Sam Houston State University
Title: On the catenary degree of elements in a numerical monoid generated by anarithmetic sequence
Abstract: We compute the catenary degree of elements contained in numericalmonoids generated by arithmetic sequences. We find that this can be done bydescribing each element in terms of the cardinality of its length set and of its
set of factorizations. As a corollary, we find for such monoids that the catenary
degree becomes fixed on large elements. This allows us to define and computethe dissonance number - the largest element with a catenary degree different fromthe fixed value. We determine the dissonance number in terms of the arithmetic
sequence’s starting point and its number of generators.
Gabriel Conant, University of Notre Dame
Title: VC-dimension in groups
Abstract: I will discuss recent work on the structure of VC-sets in groups, i.e.
subsets whose family of left translates has absolutely bounded VC-dimension. Many
tools from model theory and additive combinatorics can be adapted for “locallyamenable” VC-sets, leading to stronger structural results than for arbitrary sets in(amenable) groups. These structure results can be applied to questions concerning
sumset phenomena for VC-sets in infinite groups, as well as arithmetic regularity
for VC-sets in finite groups.
Jeshu Dastidar, Queens College (CUNY)
Title: On the maximal number of roots of a trinomial over a prime field
Abstract: Canetti, Friedlander, et al. (2002) studied the randomness of powers overfinite fields and along the way derived an analogue of Descartes rule over the finite
field Fq with q elements: They showed that the number of roots of any univariate
t-nomial, with exponents {0, a2 , . . . , at } and the differences ai − aj all relatively
prime to q − 1, is O(q (t−2)/(t−1) ). The correct optimal bounds remain a mysteryfor prime fields, even in the case of polynomials with three terms. Following thework of Kelley (2016), we expand current evidence by using a supercomputer todetermine the number of roots of these trinomials for 139, 571 ≤ p ≤ 264, 961. We
also prove that the search can be restricted to trinomials with a middle linear termwhen p − 1 has less than three distinct prime factors.
This is joint work with Viviana Márquez and Ryan Pugh under the direction ofMegan Ly and Dr. J. Maurice Rojas at the MSRI–UP REU in 2017.
Mauro Di Nasso, University of Pisa, Italy
Title: A generalized notion of asymptotic density
Abstract: Upper densities are monotone and subadditive functions from the power
set of positive integers to the unit real interval that generalize the upper densities
used in number theory, including the upper asymptotic density, the upper Banachdensity, and the upper logarithmic density. We answer a question posed by G.
Grekos in 2013, and prove the existence of translation invariant abstract upper
densities onto the unit interval, whose null sets are precisely the family of finite
sets, or the family of sequences whose series of reciprocals converge. We also showthat no such density can be atomless. (More generally, these results also hold for alarge class of “summable” ideals.) This is a joint work with Renling Jin.
Robert Donley, Queensborough Community College (CUNY)
Title: Central zeros of Clebsch-Gordan Coefficients
Abstract: Clebsch-Gordan coefficients play a fundamental role in the representation theory of SU(2) and the quantum theory of electron coupling. An alternative
approach is given using the binomial transform (Pascal’s triangle). With the associated machinery (generating functions, recurrences, Regge symmetries), we givean elementary proof of a conjecture of S. Brudno, as clarified by Rau, and noteseveral generalizations and applications.
Leonid Fel, Technion - Israel Institute of Technology
Title: Numerical semigroups generated by squares, cubes and quartics of three consecutive integers
Abstract: We derive the polynomial representations for minimal relations of generating set of numerical semigroups
Rnk = h(n − 1)k , nk , (n + 1)k i,

k = 2, 3, 4.

We find polynomial representations for degrees of syzygies in the Hilbert seriesH(z; Rnk ) of these semigroups, their Frobenius numbers F (Rnk ) and genera G(Rnk ).
We pose the conjecture about asymptotic behavior of F (Rnk ) and G(Rnk ) whenn → ∞ for arbitrary k.
Carrie Finch-Smith, Washington and Lee University
Title: Iterated Riesel and Sierpiński numbers
Abstract: A Riesel number is an odd positive integer k with the property thatk · 2n − 1 is composite for all natural numbers n. In 1956, Hans Riesel showedthat 509203 is an example of such a number, and he also showed that there areinfinitely many Riesel numbers. In this talk, we discuss iterating the process ofmultiplying by 2n and subtracting 1 while maintaining compositeness. That is, weconsider positive integers k with the property that k · 2n − 1, (k · 2n − 1) · 2n − 1,
. . . , ((· · · (k · 2n − 1) · 2n − 1) · · · ) · 2n ) − 1 are all composite for all natural numbers.
We also demonstrate analagous results for Sierpiński numbers.
Ayla Gafni, University of Rochester
Title: Additive energy and the metric Poissonian property
Abstract: Let A be a set of natural numbers. Recent work has suggested a stronglink between the additive energy of A (the number of solutions to a1 + a2 = a3 + a4
with ai ∈ A) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of A modulo 1. In particular, having “low” additive energy
forces a set to be metric Poissonian, while “high” additive energy prevents the set
from having this property. In this talk, I will discuss the history of the metric
Poissonian property and its connection to additive energy. I will then present someresults related to the intermediate range of energy, where other factors play a rolein determining if a set has the metric Poissonian property.
Daniel Glasscock, Northeastern University
Title: Combinatorial cubes and solutions to linear equations in Piatetski-Shapiro
sets
Abstract: It is a long-standing open problem to determine whether or not the setof squares contains a combinatorial cube with 7 elements, a configuration of theform {x, y, z, x + y, x + z, y + z, x + y + z}. In this talk, I will describe a metricalsolution to the problem of finding such configurations in Piatetski-Shapiro sets,
PS(α) = {bnα c | n ∈ N}. Behind the scenes is an interesting dichotomy exhibitedas the exponent α varies: for almost all small α, the set PS(α) contains solutions
to linear equations, while for almost all large α, the set PS(α) does not.
Robert Hough, SUNY Stony Brook
Title: The local limit theorem on nilpotent Lie groups
Abstract: I will discuss new local limit theorems on the Heisenberg group, and on
general connected, simply connected nilpotent Lie groups which extend the classof driving measures, generalizing earlier works of Breuillard and Alexopoulos. The
techniques used include a permutation group action, application of the GowersCauchy-Schwarz inequality, and the use of an approximate additive structure of
the large spectrum of probability measures on additive groups. The work on the
Heisenberg group is joint with Persi Diaconis.

Trevor Hyde, University of Michigan
Title: Polynomial factorization statistics and point configurations in R3
Abstract: Factorization statistics are functions defined on the set Polyd (Fq ) of degree d polynomials with coefficients in Fq that only depend on the factorization typeof a polynomial. We show there is a surprising connection between the expectedvalues of factorization statistics on Polyd (Fq ) and the structure of the cohomologyof the space of point configurations in R3 as a symmetric group representation. This
result forms a bridge between number theory and combinatorics on one hand, representation theory and topology on the other. We will demonstrate this interaction
through examples if time permits.
Brad Isaacson, New York City Tech (CUNY)
Title: On some elementary character sums
Abstract: We discuss some character sums that can be evaluated using elementary
techniques from algebra and number theory. As a corollary, we express a twisted
generalization of the classical Dedekind sum as a linear combination of generalized
Bernoulli numbers.
John H. Johnson, Ohio State University
Title: Upper Banach density along filters
Abstract: A recent observation of John Griesmer states that for upper Banachdensity d∗ (A) > α if and only if for every finite nonempty F ⊆ N we have {t ∈ N :|A ∩ (F + t)| ≥ α · |F |} 6= ∅. This observation makes precise the view that having
positive upper Banach density is the same as being a “rarefied thick set”. (A set
is thick if and only if {t ∈ N : |A ∩ (F + t)| = |F |} 6= ∅ for every finite nonempty
F ⊆ N.) Generalizing this observation, we can define a notion of upper Banachdensity relative to filters on (N, +). We show that, under certain conditions, this“filtered” notion of density still shares many of the same combinatorial properties as
the classical notion of upper Banach density such as translation invariance, partition
regularity, and recurrence. Joint work with Florian Richter.
Mizan R. Khan, Eastern Connecticut State University
Title: Hensley’s Theorem on clean lattice tetrahedra with interior lattice points
Abstract: An empty lattice tetrahedron is a tetrahedron in R3 whose vertices arelattice points, but does not contain any other lattice points. The typical example
is the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, n) where n isany non-zero integer. Since n is unbounded, this shows that there is no boundon the volume of empty tetrahedra. A clean tetrahedra is a lattice tetrahedron
whose sides do not contain any lattice points other than the vertices. (It is permissible for lattice points to lie in the interior.) An example of a clean lattice
tetrahedron which is not an empty lattice tetrahedron is the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (2, 3, 11). It is natural to extrapolate from theempty tetrahedra case that clean lattice tetrahedra containing a fixed non-zero
number of lattice points in the interior have unbounded volume. In the 80’s Hensley proved that this was untrue. He gave an inductive proof of the fact that theset of clean tetrahedra containing a fixed non-zero number of lattice in the interior
have bounded volume. We will describe the geometric idea underlying the proof.
Angel Kumchev, Towson State University
Title: A hybrid of two theorems of Piatetski-Shapiro
Abstract: The Piatetski-Shapiro primes are a classical example of a thin sequence ofprimes that has served as a test case in many investigations on additive problems inthin sets of primes. Another question about primes first posed by Piatetski-Shapiro
deals with the solubility in primes p1 , . . . , ps of the Diophantine inequality|pc1 + · · · + pcs N | < .
Here,  > 0 is small, c > 1 is non-integer, and N is a large parameter. This versionof the Waring-Goldbach problem for non-integer exponents has attracted a lot of
attention during the last quarter of a century. In this talk, I will present some joint
work with Zh. Petrov (Sofia University) on the solubility of the above Diophantine
inequality in Piatetski-Shapiro primes.
Kevin Kwan, Columbia University, and Steven J. Miller, Williams College
Title: Near-perfect, within-perfect, and order-a-abundant numbers
Abstract: A perfect number n equals the sum of its proper divisors, or σ(n) = 2nwhere σ is the sum of divisors function. Perfect numbers have been studied since
Euclid, but many ancient questions, such as the infinitude of perfect numbers andthe existence of odd perfect numbers, remain wide open. Building on the analytic
techniques of Erdős, Pollack, and Pomerance, we study two types of approximate
perfect numbers. Given a non-negative function k, we say a number n is k-withinperfect if |σ(n) − 2n| < k, and we say n is k-near-perfect if n can be written as thesum of all but at most k of its proper divisors. We prove the existence of a phasetransition in the density of k-within-perfect numbers as k varies from sub-linear
to super-linear and establish some results on their growth rate. This generalizes
previous results, which were limited to the case where k is constant. If time permits
we will mention some related results on generalizations of abundant numbers.
Steven Leth, University of Northern Colorado
Title: Some questions and results about SIM sets
Abstract: In this talk I will discuss recent joint work with Isaac Goldbring involving
the use of nonstandard methods and “SIM sets.” These are standard sets with avery natural measure property in the non-standard setting. There exist SIM setsthat have zero Banach Density, but a long-standing open question asks whether
every set of positive Banach density contains a SIM set. With the recent resolutionof the question of the existence of sumsets in sets of positive Banach density byMoreira, Richter and Robertson, it is natural to ask whether the same result istrue for SIM sets. We make some modest progress towards this goal, and also give
improved characterizations of SIM sets.

Huixi Li, Clemson University
Title: On Lemoine’s conjecture and a theorem by Estermann
Abstract: In 1931 Estermann proved that every sufficiently large integer is a sum ofa prime and a square-free number. If Goldbach’s conjecture on even numbers andLemoine’s conjecture on odd numbers are both true, then we know every integer
greater than 3 is a sum of a prime and a square-free number with at most two prime
divisors. In this presentation I will talk about the result that every sufficiently large
odd integer is a sum of a prime and a square-free integer with at most three prime
divisors. Then together with Chen’s theorem, we know every sufficiently large
integer is a sum of a prime and a square-free number with at most three prime
divisors.
Jared Duker Lichtman, Dartmouth College
Title: Explicit estimates for the distribution of numbers free of large prime factors
Abstract: A number is called y-smooth if all its prime factors are at most y. Thereis a large literature on the distribution of y-smooth numbers up to x. Asymptotic
estimates for this distribution are quite useful in many applications, including theanalysis of factorization and discrete logarithm algorithms. However, very little
is known about this distribution that is numerically explicit. We generalize thesaddle-point method of Hildebrand and Tenenbaum, giving explicit and fairly tightintervals in which the true count lies. For example, we determine that the number of
500-digit integers free of ≥ 35-digit prime factors lies within 0.65% of 1.505 × 10482 .
This interval is so tight we can exclude the famous Dickmande Bruijn asymptotic
estimate as too small (1.472 × 10482 ) and the HildebrandTenenbaum main term as
too large (1.513 × 10482 ). One of the essential ideas in the proof is to relate thedesired count of smooth numbers to a contour integral over a vertical line in thecomplex plane. The integrand includes an oscillatory sum over the primes, which
requires an explicit form of the Prime Number Theorem. Joint work with Carl
Pomerance.
Diego Marques, Universidade de Brası́lia, Brazil
Title: On some problems proposed by Kurt Mahler
Abstract: In this lecture, we will present some problems proposed by K. Mahler
and related to the arithmetic behavior of transcendental functions. Moreover, weshall provide the recent advances on this topic.
Ariane Masuda, New York City Tech (CUNY)
Title: Permutation polynomials over Fq2 from rational functions
Abstract: Let Fq be the finite field of q elements. A polynomial f in Fq [x] is apermutation polynomial over Fq if the induced mapping x 7→ f (x) is a bijection in
Fq . We discuss a method for constructing permutation polynomials over Fq2 by
using rational functions that induce bijections either on the set µq+1 of the (q + 1)th roots of unity or between µq+1 and Fq ∪ {∞}. This is joint work with Daniele
Bartoli and Luciane Quoos.

Azita Mayeli, Queensborough CC and CUNY Graduate Center
Title: Existence of no Gabor orthogonal bases on the smooth and symmetric convex
sets
Abstract: Let K be a convex and symmetric bounded set in Rd , d ≥ 2, withsmooth boundary. Using a combinatorial approach, in this talk we show that for
d 6= 1(mod4), the indicator function of K can not serve as an orthogonal Gaborwindow function for L2 (Rd ), i.e., there is no countable set S ⊂ R2d such that theGabor family G(1K , S) = {e2πix·b 1K (x − a) : (a, b) ∈ S} is an orthogonal basis forL2 (Rd ). This is joint work with Alex Iosevich.
Katie McKeon, Rutgers University
Title: Low-lying fundamental geodesics in an arithmetic hyperbolic 3-manifold
Abstract: We’ll examine closed geodesics in the quotient of hyperbolic three spaceby the discrete group of isometries SL(2,Z[i]). There is a correspondence between
closed geodesics in the manifold, the complex continued fractions originally studied
by Hurwitz, and binary quadratic forms over the Gaussian integers. According
to this correspondence, a geodesic is called fundamental i f the associated binary
quadratic form is. Using techniques from sieve theory, symbolic dynamics, and thetheory of expander graphs, we show the existence of a compact set in the manifold
containing infinitely many fundamental geodesics.
Nathan McNew, Towson University
Title: Counting primitive subsets of {1, 2, ...n}Abstract: Let f (n) count the number of primitive subsets of the integers {1, 2, ...n},where primitive means that no integer in the set divides another√ integer in the set.
n
In 1988 Cameron and Erdős noted that 2n > f (n) > 2b 2 c ≈ ( 2)n , which follows
from the fact that any subset of the integers from dn/2e to n is primitive. They
conjectured that in fact f (n) was asymptotic to cn for some constant c, and showedthat 1.55n < f (n) < 1.59n . In 2017 this conjecture was proven by Angelo, whoshowed that f (n) ∼ cn for some c, but the proof is ineffective in the sense that itgives no information about the value of the constant c.
We give an alternate proof of this result which gives more than just an asymptotic
for f (n) and also shows that the constant c is effectively computable. For example
we can improve the bounds to 1.572 < c < 1.574. We also show that the same
method can be used to count subsets which are geometric-progression-free.
Joel Moreira, Northwestern University
Title: A solution to the Erdős sumset conjecture
Abstract: Erdős conjectured that every subset A of the natural numbers withpositive density contains a sumset B + C := {b + c : b ∈ B, c ∈ C} for infinite
sets B and C of natural numbers. I will present a solution of this conjecture,obtained recently in joint work with Richter and Robertson. There are three main
steps. The first is to rewrite the statement using functions instead of sets; we dothis using the language of ultrafilters. The second is to decompose (the indicatorfunction of) a set with positive density into an ”almost periodic” component and a”pseudo-random” component. In fact, we need two distinct such decompositions.

Finally we tie everything together, using some ideas and tools from ergodic theory.
No knowledge of ergodic theory or ultrafilters will be assumed.
Mel Nathanson, CUNY (Lehman College and the Graduate Center)
Title: Explicit computation of Sinkhorn limits
Abstract: Let A be an n×n positive matrix, and let (Ak )∞k=1 be the sequence of positive matrices obtained by alternately column scaling and row scaling. Sinkhorn’s
theorem asserts that the sequence (Ak )∞k=1 converges to a doubly stochastic matrix.
Explicit limits are computed for 3 × 3 symmetric matrices.
Kevin O’Bryant, College of Staten Island (CUNY)
Title: Explicit bounds for primes in arithmetic progressions
Abstract: We present fully explicit bounds on the number of primes less than x
and congruent to a mod q, valid for all a, q, and modestly sized x. Joint work with
Mike Bennett, Greg Martin, and Andrew Rechnitzer.
Péter Pál Pach,
Title: On multiplicative Sidon sets
Abstract: While the topic of additive Sidon sets is well-studied in additive number
theory, much less attention has been devoted to the multiplicative case. In this talkwe will discuss results about multiplicative Sidon sets.
Ram Krishna Pandey, Indian Institute of Technology Roorkee, India
Title: Direct and inverse theorems on the general h-fold sumsets and on the sumsof dilated sets
Abstract: Let A be a nonempty finite set of integers and h and r be positive integers.
The generalized h-fold sumset, denoted by h(r) A, is the sum of h elements of A,
where each element appears in the sum can have at most r repetitions. Further,
the Minkowski sum of the dilated sets is defined by
m1 · A + m2 · A + · · · + mk · A = {m1 a1 + m2 a2 + · · · + mk ak ; ai ∈ A for 1 ≤ i ≤ k}.
Here, m1 , m2 , . . . , mk are positive integers with gcd(m1 , m2 , . . . , mk ) = 1. In thistalk, we present some direct and inverse results for |h(r) A| and |m1 · A + m2 · A +· · · + mk · A|.
Cosmin Pohoata, California Institute of Technology
Title: Higher energies in additive combinatorics and discrete geometry
Abstract: Using geometric variants of additive higher moment energies (introducedby Schoen and Shkredov), we discuss improved bounds for the problem of distinct
distances with local properties and new bounds for problems involving expanding
polynomials in R[x, y] when one or two of the sets have structure.

Yuneid Puig de Dios, University of California, Riverside
Title: Recurrence sets of linear operators with high piecewise syndeticity level.
Abstract: Our talk is concerned with the interplay of dynamics of linear operatorsand combinatorial number theory. We study the dynamics of linear and bounded
operators with recurrence sets having a high piecewise syndeticity level, and show aresult motivated by a question posed by Sophie Grivaux concerning the regularityof the orbits of frequently hypercyclic operators.

Alex Rice, Millsaps College
Title: Extending the best known bounds for the Furstenberg-Sárközy Theorem
Abstract: It is a well known result, established independently by Sárközy and
Furstenberg, that a set of integers with positive upper density must contain two
distinct elements that differ by a perfect square. The best-known quantitative
upper bounds for this result were established with an intricate Fourier analytic
argument by Pintz, Steiger, and Szemerédi. In this talk, we discuss the extensionof these bounds from perfect squares to, at long last, the largest possible class ofpolynomials, as well as other related results.
David Ross, University of Hawaii
Title: Non-archimedean extensions of R and diophantine equations connected to
Egyptian fractions
Abstract: Egyptian fractions are closely related to a variety of interesting Diophantine equations, such as the Kellogg Equation (1/x1 + 1/x2 + · · · + 1/xs = 1),
the Znám Equation (1/x1 + · · · + 1/xs + 1/x1 x2 · · · xs = a), and the Lagarias
Equation (c(1/x1 + · · · + 1/xs ) + b/x1 x2 · · · xs = a). I’ll discuss the way certain
non-Archimedean extensions of R can give insight into the structure of the solution
sets of such equations, and therefore into the set of numbers representable as Egyptian fractions. For example, I’ll show that sets such as these are generally closed,
extending an old result of Sierpinski.
Eric Rowland, Hofstra University
Title: Enumeration of binomial coefficients by their p-adic valuations
Abstract: In 1947 Nathan
 Fine obtained a beautiful formula for the number ofnbinomial coefficients m, for fixed n, that are not divisible by p: if the standard
base-p representation of n is n` · · · n1 n0 , then this number is(n0 + 1)(n1 + 1) · · · (n` + 1).
Subsequently, many authors found formulas counting binomial coefficients with pn
adic valuation νp ( m) = α for particular values of p and α, but a general formularemained elusive. We give a matrix product, generalizing Fine’s result, for thegenerating function
Tp (n, x) :=

n
X
m=0

n

xνp ((m))

which simultaneously counts binomial coefficients with p-adic valuation α for allα ≥ 0. Namely, if
d+1 p−d−1
Mp (d) :=dx(p − d) xthen
Tp (n, x) = 1

 
1
0 Mp (n0 ) Mp (n1 ) · · · Mp (n` ).
0

A further generalization, involving k × k matrices, counts multinomial coefficientsby their p-adic valuations.
Andrey Rukhin, NSWCDD. Dept. Of Navy
Title: Alternative proofs of Steiner’s 1-cycle theorem
Abstract: Within the context of the 3x + 1 Problem, Steiner’s 1-Cycle Theorem is a
result pertaining to the non-existence of certain periodic orbits (known as circuits)
within the 3x + 1 dynamical system. For all a, b ∈ N, Steiner demonstrated thatrational expressions of the form
2a − 12a+b − 3b
do not assume a positive integer value, except in the case where a = b = 1 (corresponding to the (1, 4, 2) orbit); he does so by appealing to the continued fraction
expression of log2 3, transcendental number theory, and numerical computation.
In this talk, the speaker offers alternative proofs of Steiner’s theorem from a
algebraic perspective. Assuming an upper bound on periodic iterates that was established by Belaga and Mignotte (appealing to the work of Baker and Wüstholz),
these proofs employ elementary identities from modular arithmetic, weighted binomial coefficients, and the graded 2-adic and 3-adic expansions of rational expressions
such as above.
John R. Schmitt, Middlebury College
Title: Distinct Partial Sums in Cyclic Groups
We consider some simply stated problems and conjectures arising from the studyof combinatorial designs. These problems may be broadly described as follows: afinite subset of the elements of some group is given and one wishes to order theelements of this finite subset so that the sequence of partial sums has terms thatare distinct. Within this setting there are a plethora of questions that one might
consider; these arise as one varies the group, places restrictions on the elements ofthe subset chosen, or imposes additional restrictions upon the sequence of partial
sums beyond the terms being distinct.
To be specific, let (G, +) be an abelian group and consider a subset A ⊆ Gwith |A| = k. Given an ordering (a1 , . . . , ak ) of the elements of A, define its partial
Pjsums by s0 = 0 and sj = i=1 ai for 1 ≤ j ≤ k.
We will consider the following conjecture attributed to Alspach: For any cyclic
group Zn and any subset A ⊆ Zn \ {0} with sk 6= 0, it is possible to find anordering of the elements of A such that no two of its partial sums si and sj areequal for 0 ≤ i < j ≤ k.
We also consider a weakening of this conjecture due to Archdeacon.

We address these conjectures in the case that n is prime and do the following.
We show how Noga Alon’s Combinatorial Nullstellensatz can be used to frame these
conjectures. Further, in the case that n is prime, we verify computationally thatthe conjecture is true for small values of |A|. In the case that n is prime, we showthat a sequence of length k having
distinct partial sums exists in any subset of√
Zn \ {0} of size at least 2k − 8k in all but at most a bounded number of cases.
Joint work with Jacob Hicks and Matt A. Ollis.
Steven Senger, Missouri State University
Title: Rainbow point configurations
Abstract: We prove some very general results about which colorings of sets guarantee special subsets with each element from a different color class. We present
many examples such as point configurations in various vector spaces, as well assome additive combinatorial type results in abelian groups.
Arseniy Sheydvasser, Yale University
Title: Rigidity in the Ulam sequence
Abstract: Let U (a, b) denote the generalized Ulam sequence starting with integers
a, b, such that each subsequent term is the smallest integer that can be written
as the sum of two distinct preceding terms in exactly one way. Ulam, for which
the sequence is named, introduced U (1, 2) in 1964. Since then, it has continued toengender both interest and frustration due to appearing to satisfy clear patterns
that nevertheless continue to be impossible to prove. In this talk, we shall discuss
some more recent results about families of Ulam sequences. Our general themeis that while any individual Ulam sequence is chaotic, families of Ulam sequences
are surprisingly rigid. Along the way, we shall discuss both some of the uses andlimitations of model theory in studying this problem.
Satyanand Singh, New York City Tech (CUNY)
Title: A Refinement of the Calkin Wilf Tree with some of its Implications
Abstract: We study a refinement of the Calkin-Wilf tree due to Nathanson in thispresentation. In particular,
weof such trees associated consider the properties

1 u
1 0
with the matrices Lu =and Rv =, where u and v are nonnegative
0 1
v 1
integers. We extend several known results of the original Calkin-Wilf tree, including
symmetry, numerator-denominator, and successor formulas, to this new setting.
The talk will columinate with some recent results on maximal entries in certain
matrix monoids. This is based on joint work with Sandie Han, Ariane M. Masuda
and Johann Thiel.
Christoph Spiegel, Universitat Politécnica de Catalunya, Spain
Title: Going beyond 2.4 in Freiman’s 2.4-Theorem
Abstract: Freiman’s 2.4-Theorem states that any set A ⊂ Zp satisfying |2A| ≤2.4|A| − 3 and |A| < p/35 can be covered by an arithmetic progression of length at
most |2A| − |A| + 1. A more general result of Green and Ruzsa implies that thiscovering property holds for any set satisfying |2A| ≤ 3|A| − 4 as long as the ratherstrong density requirement |A| < p/10215 is satisfied. We present a version of thisstatement that allows for sets satisfying |2A| ≤ 2.48|A| − 7 with the more modest
density requirement of |A| < p/1010 .
This is joint work with Pablo Candela and Oriol Serra.
Wenbo Sun, Ohio State University
Title: Solution counting problem and quantitative recurrence theorem
Abstract: Let p(x1 , . . . , xn ) be a linear function with n variables. In combinatorics,
the solution counting problem studies the number of solutions for the equationp(x1 , . . . , xn ) = 0 for x1 , . . . , xn lying in a given subset of integers. In ergodic theory,the quantitative recurrence theorem studies the lower bound for the measure of theset A ∩ T a1 n A ∩ · · · ∩ T ad n A over a large set of return time n for some integers
a1 , . . . , ad . In this talk, I will talk about recent advances for these topics, anddiscuss the connection between these two questions.
This is joint work with Sebastian Donoso, Anh Leeand Joel Moreira.
Caroline Amelia Terry, University of Maryland
Title: A stable arithmetic regularity lemma in finite abelian groups
Abstract: The arithmetic regularity lemma for Fnp (first proved by Green in 2005)
states that given A ⊆ Fnp , there exists H ≤ Fnp of bounded index such that A is
Fourier-uniform with respect to almost all cosets of H. In general, the growth of theindex of H is required to be of tower type depending on the degree of uniformity,
and must also allow for a small number of non-uniform elements. Previously, in joint
work with Wolf, we showed that under a natural stability theoretic assumption, thebad bounds and non-uniform elements are not necessary. In this talk, we present
results extending these results to stable subsets of arbitrary finite abelian groups.
This is joint work with Julia Wolf.
Lola Thompson, Oberlin College
Title: Divisor-sum fibers
Abstract: Let s(·) denote the sum-of-proper-divisors function, that is, s(n) =P
d|n, d<n d. Erdős–Granville–Pomerance–Spiro conjectured that, for any set A
of asymptotic density zero, the preimage set s−1 (A) also has density zero. We
prove a weak form of this conjecture. In particular, we show that the EGPS con1
jecture holds for infinite sets with counting function O(x 2 +(x) ). We also disprovea hypothesis from the same paper of EGPS by showing that for any positive numbers α and , there are integers n with arbitrarily many s-preimages lying betweenα(1 − )n and α(1 + )n. This talk is based on joint work with Paul Pollack andCarl Pomerance.
Henry Towsner, University of Pennsylvania
Title: Generalizing VC dimension to higher arity
Abstract: The notion of bounded VC dimension is a property at the intersectionof combinatorics and probability. This family has been discovered repeatedly andstudied from various perspectives - for instance, in model theory, theories withbounded VC dimension are known as NIP (the theories which do Not have the Independence Property). One useful property is that graphs with bounded VC dimension are the graphs that can be always be finitely approximated in a ”random-free”
way: graphs with bounded VC dimension satisfy a strengthening of Szemeredi’s
Regularity Lemma in which the densities between the pieces of the partition are
either close to 0 or close to 1. The generalization of VC dimension to higher arity,
known in model theory as k-NIP for various k, has been less well-studied. We summarize some known facts about this generalization, including a new result (joint
with Chernikov) showing k-NIP hypergraphs have a similar kind of approximation
with only ”lower order” randomness.
Yuri Tschinkel, Courant Institute, New York University
Title: Arithmetic of del Pezzo surfaces
Abstract: I will survey recent advances on problems of rationality and rational
points on del Pezzo surfaces.
Maximilian Wötzel, Universitat Politècnica de Catalunya, Spain
Title: Set systems with distinct sumsets
Abstract: A family A of k-subsets of {1, 2, . . . , N } is a Sidon system if the sumsets
A + A0 , A, A0 ∈ A are pairwise distinct. We show that the largest
 cardinality−1
Fk (N ) of a Sidon system of k-subsets of [N ] satisfies Fk (N ) ≤ N
k−1 + N − k andthe asymptotic lower bound Fk (N ) = Ωk (N k−1 ). More precise bounds on Fk (N )are obtained for k ≤ 3. We also obtain the threshold probability for a random
system to be Sidon for k ≤ 3. This is joint work with Javier Cilleruelo and Oriol
Serra.