CANT 2014 Abstracts
New York Number Theory Seminar
Twelfth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 27-30, 2014

Principal lecturer:
Harald HelfgottÉcole Normale Supérieure, Paris

Lecture 1: “The ternary Goldbach problem”
Abstract: The ternary Goldbach conjecture (1742) asserts that every odd number
greater than 5 can be written as the sum of three prime numbers. Following thepioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every
odd number larger than a constant C satisfies the conjecture. In the years sincethen, there has been a succession of results reducing C, but only to levels much toohigh for a verification by computer up to C to be possible (C > 101300 ). (Worksby Ramare and Tao have solved the corresponding problems for six and five prime
numbers instead of three.) My recent work proves the conjecture. We will go overthe main ideas in the proof.
Lecture 2. “Major arcs in the solution to the ternary Goldbach problem”
Abstract: The ternary Goldbach conjecture states that every oddPnumber n ≥ 7
is the sum of three primes. The estimation of sums of the form p e(αp)η(p/x),
where η is a weight, has been a central part of the main approach to the conjecturesince Hardy, Littlewood and Vinogradov. We will see how to estimate such Fourier
series for α in the so-called major arcs, i.e., for α close to a rational of small denominator. In the process, we will discuss estimates on parabolic cylinder functions
that will make it possible to use weights based on the Gaussian in future explicit
work in analytic number theory.
Lecture 3. “Minor arcs in the solution to the ternary Goldbach problem”
Abstract: The ternary Goldbach conjecture states that every oddPnumber n ≥ 7
is the sum of three primes. The estimation of sums of the form p e(αp)η(p/x),
where η is a weight, has been a central part of the main approach to the conjecture since Hardy, Littlewood and Vinogradov. We will see how to estimate such
Fourier series for α in the so-called minor arcs, i.e., for α not close to any rationalsof small denominator. This is the technical heart of the speaker’s recent proof of
the ternary Goldbach conjecture. There are several qualitative improvements toprevious bounds. In particular, we shall examine general ideas for reducing the1

cost of Vaughan’s identity, as well as a way to exploit the tails of minor arcs in thecontext of the large sieve.
Lecture 4. “Growth in groups: ideas and perspectives”
Abstract: This will be a survey of methods developed in the last decade to prove
results on growth in non-commutative groups. These techniques have their roots
in both additive combinatorics, group theory and other fields. We discuss linear
algebraic groups, with SL2 (Z/pZ) as the basic example, as well as permutationgroups. There remain several outstanding open problems, some of them algorithmic
in nature.

Invited lectures
Sukumar Das Adhikari, Harish-Chandra Research Institute, Allahabad, India“Generalizations of certain zero-sum theorems: Some recent progress”
Abstract: Consider a finite abelian group G (written additively). A sequence S =g1 · . . . · gl over G is called a zero-sum sequence if g1 + · · · + gl = 0, where 0 is theidentity element of the group. Inspired by a well known result of Erdős, Ginzburg
and Ziv, the area of zero-sum theorems in combinatorial number theory has seen
a rapid growth in the recent years. After a brief introduction to the early resultsin the area, we proceed to discuss some recent results related to their weighted
generalizations.

Paul Baginski, Fairfield University“Factoring within arithmetic progressions, revisited”
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 subsemigroups of N, their factorization properties
differ greatly from the unique factorization one enjoys in N. At CANT 2011, I
gave a survey of the then-known factorization properties of these semigroups. Most
critically, they are not Krull monoids and thus do not have a class group which
fully captures the factorization behavior. Nonetheless, a certain finite abelian groupassociated to it does convey some of the information one would expect from a classgroup. I will recount some of the more peculiar factorization properties of ACMs
from before, to give a flavor of the unexpected behavior of these simple-looking
monoids. But the majority of the talk will concentrate on the particular propertyof elasticity, on which there has been significant recent progress, all closely tied to
combinatorial configurations within this ”class group.”Thomas Bloom, University of Bristol
Title: “Quantitative improvements for Roth’s theorem”
Abstract: We prove a new structural result for the set of large Fourier coefficientsand show that for some problems it can substitute as a quantitatively superiorversion of the well-known Chang’s lemma. In particular, we use it to show that
if A ⊂ {1, . . . , N } contains no non-trivial three term arithmetic progressions then|A| ≤ (log log N )4 / log N .

Bren Cavallo, CUNY Graduate Center”The subset sum problem and the conjugacy problem in polycyclic groups”
Abstract: In this talk we construct polycyclic groups Gn whose conjugacy problem
is at least as hard as the subset sum problem with n indeterminates. We further
prove that the conjugacy problem over the groups Gn is NP-complete where theparameters of the problem are taken in terms of n and the length of the elements
given on input. (Joint work with Delaram Kahrobaei)

Alan Chang, Princeton University“Newman’s conjecture in various settings”
Abstract: Polya introduced a deformation of the Riemann zeta function ζ(s), andDe Bruijn and Newman found a real constant Λ which encodes the movement of thezeros of ζ(s) under the deformation. The Riemann hypothesis (RH) is equivalentto Λ ≤ 0. Newman conjectured that Λ ≥ 0, and remarked that “the new conjecture
is a quantitative version of the dictum that the Riemann hypothesis, if true, is only
barely so.” We generalize the conjecture and the techniques to apply to a widerclass of L-functions, including automorphic L-functions as well as function fieldL-functions. Each “family” of function field quadratic L-functions gives a different
version of Newman’s conjecture. The recent proof of Sato-Tate for elliptic curves
over totally real fields allows us to prove a version of Newman’s conjecture involving
fixed D ∈ Z[T ] of degree 3.

Jean-Marc Deshouillers, IPB-IMB Bordeaux, France“Sums of 3 or 4 cubes: Heuristic and numerical approaches”
Abstract: A frustrating question in additive number theory is that we are notable to decide whether sums of 3 cubes have a positive density or whether any
sufficiently large integer is a sum of 4 cubes. In 1960, Erdős and Rényi developed aprobabilistic model for s-th powers and studied the sum of s such elements. Goguelmade their work more precise some 15 years later, and Hennecart, Landreau and
Deshouillers developed a model taking into account the arithmetic properties of s-th
powers which was ignored in the Erdős-Rény model, and showed the experimental
adequacy of their model. After a quick review of those results, a recent workof Cilleruelo and Deshouillers will be presented, showing that in the Erdős-Rény
model, sums of s-powers have no bounded gaps (a precise estimation of the larger
gaps will be given), although they have a positive density.

Charles Helou, Penn State Brandywine“Some functions related to the general Erdős-Turán conjecture”
Abstract: The general Erdős-Turán conjecture states that if A = {a1 < a2 <· · · < an < . . . } is an infinite sequence of non-negative integers satisfying an ≤ cn2
for all n ≥ 1, with a constant c > 0, then the number of representations of theintegers as sums of two elements of A is unbounded. We study the functions
ψ, ψ , and ψ − defined on the sets L(n), L (n), and L− (n) of finite sequences
A = {a1 < a2 < · · · < an } of n elements satisfying ak ≤ k 2 , ak ≤ k 2− (0 <  < 1),
and ak < k 2 , respectively, giving the minimum over these sets of the maximal
number of representations of integers by such sequences. We give an equivalent
formulation of the general Erdős-Turán conjecture in terms of the first function,and some computational results about it, and we state a sufficient condition forthis conjecture involving the last two functions. (Joint work with G. Grekos, L.
Haddad and J. Pihko.)Nathan Kaplan, Yale University”Arcs in the projective plane”
Abstract: A (k, n)-arc of P2 (Fq ) is a set of n rational points in the projective plane,
no more than k of which lie on a line. Segre’s famous theorem on (2, n)-arcs saysthat when q is odd, the largest (2, n)-arc is of size q +1 and every such arc is the zero
set of a smooth conic. When q is even there are (2, q + 2)-arcs called hyperovals,
which are yet to be completely classified.
In addition to asking for the largest size of an arc, we can ask for the numberof arcs of a given size. We will explain how counting arcs with a small number ofpoints is related to counting certain interesting surfaces with many rational points.
We will also discuss approaches to constructing large (k, n)-arcs based on propertiesof curves over finite fields.

Sandra Kingan, Brooklyn College (CUNY)“Splitters and decomposers for binary matroids”
Abstract: Let M denote the class of binary matroids with no minors isomorphic to
M1 , . . . , Mk . A splitter N for M is a 3-connected matroid such that no 3-connectedmatroid in M has a proper N -minor. A 3-decomposer N for M is a 3-connectedmatroid with a non-minimal exact 3-separation (A, B) such that any matroid M
in M with an N -minor has a 3-separation (X, Y ) such that A ⊆ X and B ⊆ Y .
Splitters and 3-decomposers capture the structure present in excluded minor classes.
We characterize the class of binary matroids with no minors isomorphic to S10 or∗by identifying the splitters and 3-decomposers for this class. The matroid S10 isS10a prominent binary matroid and several recent theorems on internally 4-connectedmatroids can be obtained as corollaries of our characterization.

Angel V. Kumchev, Towson University“New results in the Waring-Goldbach problem”
Abstract: A series of recent breakthroughs by T.D. Wooley in the study of Vinogradov’s mean-value integral has led to improved bounds for Weyl sums over primes.
In this talk, I will announce some applications of the new bounds to the WaringGoldbach problem in eight or more variables. (Joint work with T.D. Wooley)

Thai Hoang Le, University of Texas“Equidistribution of polynomial sequences in function fields”
Abstract: We study a function field analog of Weyl’s classical theorem on theequidistribution of polynomial sequences. Our result covers the case when the degree of the polynomial is greater than or equal to the characteristic of the field,
which has been known since Carlitz (1952) to be a natural barrier to the Weyl
differencing process in this setting. If time permits we will talk about some combinatorial applications of our result. (Joint work with Yu-Ru Liu)

Eshita Mazumdar, Harish-Chandra Research Institute, Allahabad, India“Modification of a certain method in the study of a zero-sum constant”
Abstract: For a finite abelian group G with exp(G) = n, the arithmetical invariantsmn (G) is defined to be the least integer k such that any sequence S with length kof elements in G has a zero-sum subsequence of length mn. When m = 1, it is theErdős-Ginzburg-Ziv constant and is denoted by s(G). There are weighted versionsof these constants. Here,we see a modification of a method of Griffiths which had
been used to attack a problem for some weighted version of the constant.

Nathan McNew, Dartmouth College“Random multiplicative walks and the most popular large prime divisors”
Abstract: How many multiplications should we expect to make until a product ofrandom residues modulo N is 0 modulo N ? Is it significantly more than the largest
prime factor? Which primes are we most likely to encounter as this largest prime
factor for N in the range [1, X]? Will every prime be the most popular largest
prime divisor in this interval for some value of X?

Steven J. Miller, Williams College“Continued fraction digit averages and Maclaurin’s inequalities”
Abstract: A classical result of Khinchin says that for almost all real numbers α, thegeometric mean of the first n digits ai (α) in the continued fraction expansion of α
converges to a number K = 2.6854520 . . . (Khinchin’s constant) as n → ∞. On theother hand, for almost all α the arithmetic mean of the first n continued fraction
digits ai (α) approaches infinity as n → ∞. There is a sequence of refinements of
the AM-GM inequality, Maclaurin’s inequalities, relating the 1/k th powers of thek th elementary symmetric means of n numbers for 1 ≤ k ≤ n. On the left end(when k = n) we have the geometric mean, and on the right end (k = 1) we havethe arithmetic mean. We analyze what happens to the means of continued fraction
digits of a typical real number in the limit as one moves f (n) steps away from either
extreme. We prove sufficient conditions on f (n) to ensure to ensure divergence when
one moves f (n) steps away from the arithmetic mean and convergence when one
moves f (n) steps away from the geometric mean. For typical α we conjecture thebehavior for f (n) = cn, 0 < c < 1. We also study the limiting behavior of suchmeans for quadratic irrational α, providing rigorous results, as well as numericallysupported conjectures. (Joint with Francesco Cellarosi (UIUC) and Jake Wellens(Caltech).)Mel Nathanson, Lehman College, CUNY
“A forest of linear fractional transformations”
Abstract: The Calkin-Wilf tree is an infinite binary tree whose vertices are thepositive rational numbers. Each number occurs in the tree exactly once and in theform a/b, where are a and b are relatively prime positive integers. It is possible toconstruct an analogous tree of positive linear fractional transformations of determinant 1, and to prove that this tree possesses the basic properties of the Calkin-Wilftree of positive rational numbers.

Lan Nguyen, University of Wisconsin-Parkside“On the Hasse principle for systems of binary cubic forms”
Abstract: Selmer and others have shown that cubic forms do not satisfy the Hasse
Principle. For system of cubic forms, the problem concerning the Hasse Principle
has been studied by a number of people, particularly the diagonalizable system ofsuch forms with large number of variables. For systems of cubic forms which areeither not necessarily diagonalizable or have fewer number of variables, not much
is known. In the case of binary cubic forms, it is known that any one binary cubic
form with rational coefficients satisfies the Hasse Principle. That is, it has solutions
in all p-adic completions if and only if it has rational solutions. In this talk, weshow that the Hasse Principle is satisfied for any system of binary cubic forms withrational coefficients. Our method also makes it possible to produce a new proof of
Selmer’s famous counterexample to the Hasse Principle for cubic forms, which is
independent of the Finite Basis theorem of Selmer.

Kevin O’Bryant, College of Staten Island, CUNY
“Sets without geometric progressions”
Abstract: In a CANT 2013 problem session, I raised the problem of bounding thesize of sets of positive integers that do not have subsets of the form {a, ar, ar2 , . . . , ark−1 },except for the r = 1 triviality. A flurry of progress has happened in the past year
providing sharp bounds and connections to other Ramsey-ish problems, and I will
present the highlights. I will present work of McNew, Nathanson, O’Bryant, andothers.

Alberto Perelli, University of Genova, Italy“The Selberg class of L-functions: A survey”
Abstract: Selberg introduced a general class S of L-functions defined by a set ofanalytic axioms, and proposed several interesting conjectures. Roughly speaking,the main problem in the Selberg class theory is the general converse theorem, i.e.
the characterization of the functions of given degree in the class S. In the lecture
we introduce the basic concepts of the Selberg class and describe the state of theart. The main tool for such a characterization are the nonlinear twists; the studyof these twists gives new results also in the case of the classical L-functions.

Giorgis Petridis, University of Rochester“On a question of Bukh on sums of dilates”
Abstract: Let A be a finite non-empty set in a commutative group and set
A + 2.A = {a + 2b : a, b ∈ A} and A + A = {a + b : a, b ∈ A} .
We look to bound efficiently |A + 2.A| in terms of |A| and |A + A|. That is, assume|A + A| ≤ α|A| and bound |A + 2.A| in terms of α and |A|. There is an “easy”
bound that follows from the inclusion A + 2.A ⊆ A + A + A. Plünnecke’s inequality
implies |A + A + A| ≤ α3 |A| and so |A + 2.A| ≤ α3 |A|.
Bukh asked in 2007 whether there exists p < 3 such that
|A + A| ≤ α|A| =⇒ |A + 2.A| ≤ αp |A| .
We answer the question to the affirmative by showing that one may take p =5.
2Luciane Quoos, Instituto de Matemática, UFRJ, Rio de Janeiro, Brasil“Weierstrass semigroups and algebraic geometric codes on towers of function fields”
Abstract: For applications in algebraic geometric codes, an explicit description ofbases of Riemann-Roch spaces of divisors on function fields over finite fields isneeded. We give an algorithm to compute such bases for one point divisors, andWeierstrass semigroups over an optimal tower of function fields. We also explicitly
compute Weierstrass semigroups till level eight.
We consider the optimal tower T = (Tj )j≥0 over the finite field fp2 in odd
characteristic. This tower is defined recursively by T0 = fp2 (x0 ) and, for j ≥0, Tj+1 = Tj (xj+1 ), where the function xj+1 satisfies the relation:
x2j+1 =

x2j + 1.
2xj

j
Let P∞
be the unique pole of the function x0 in Tj . For each s ∈ N fix the divisorsjsP∞ , and consider
j
j
L(sP∞
) = {z ∈ Tj | the divisor of z satisfies (z) ≥ −sP∞},

the Riemann-Roch space. The main result is an algorithm to compute such basesjand obtain, at the same time, the Weierstrass semigroup H(P∞) at level j. (Joint
work with Francesco Noseda (UFRJ/Brazil) and Gilvan Oliveira (UFES/Brazil))

Steven Senger, University of Delaware“Some Erdős-type dot product problems and applications”
Abstract: Erdős asked many questions about the distances determined by large
finite point sets. Here, we consider variants of these questions concerning dot
products (also known as scalar products or inner products) instead of distances. We
will survey results, some known and some new, as well as applications, specificallyto additive number theory.

Satyanand Singh, New York City Tech (CUNY)“Expectations of random h-fold sums of certain uniform distributions”
Abstract: Let U1 , U2 ,... be independent random variables, each uniformly distributed on (0, 1). For 0 < x ≤ 1 and α a fixed constant, where α ∈ (0, 1], we definePhhx,α to be the least h such that k=1 Uk α > x . We will find the expectation ofhx,α for specific values of α. It is known that the expected value of h1,1 = e and thiswas extended in a Monthly problem proposed by Shai Covo to find the expectationof hx,1/2 . We will derive closed form expressions for additional values of α anddiscuss related open problems. (Joint work with my student Steven Tipton)

Jonathan Sondow, New York
“Primary pseudo-perfect numbers and the Erdős-Moser diophantine equation”
Abstract: A primary pseudo-perfect number (PPN for short) is an integer K > 1
that satisfies the Egyptian fraction equation
X1
1
+
= 1,
K
p
p|K

where p denotes a prime. I review the known results on PPNs and mention connections with perfectly weighted graphs, singularities in complex dimension 2, Curtiss’s
solution to Kellogg’s problem, Sylvester’s sequence, Giuga numbers, and Znám’s
problem. Then I make some elementary but intriguing observations about PPNs
and on this basis pose some conjectures. Assuming one of them, I give a conditional proof of a new record lower bound on non-trivial solutions to the Erdős-Moser
Diophantine equation
1n + 2n + · · · + (k − 1)n + k n = (k + 1)n .
(Joint work with Kieren MacMillan. Our paper is being revised for the Monthly.)

Yonutz V. Stanchescu, Afeka Tel Aviv Academic College and The Open University of Israel“Structure theorems for dilates of integers and small doubling sets in BaumslagSolitar groups”
Abstract: We present some new direct and inverse results concerning Minkowski
sums of dilates of integers. We also investigate small doubling problems in BaumslagSolitar groups. In order to obtain these results, we establish a new connection
between abelian sumsets estimates and growth in non-abelian groups. (Joint workwith G. A. Freiman, M. Herzog, P. Longobardi and M.Maj)

Tim Susse, CUNY Graduate Center and University of Nebraska“Commutators, quasipolynomials and lattice points”
Abstract: In this talk we will survey some recent results from Geometric Group
Theory that use Number Theory and have some surprising Number Theoretic patterns. We will first define stable commutator length (scl), an interesting quantity
in Geometric Group Theory, and describe several results where the proofs hinge onunderstanding convex hulls of lattice points. We will then restrict our attention tofree products of cyclic groups and show that scl is a quasirational function in theorders of the free factors.

Johann Thiel, New York City Tech (CUNY)“RATS Sequences in General Bases”
Abstract: Conway’s RATS sequences are generated by repeating the following process: Begin with a positive integer, reverse the order of the digits, add the original
integer to its reverse, then sort the sum’s digits in increasing order from left toright. We consider this process for integers written in other bases besides 10 anddiscuss the long-term behavior of such sequences. In particular, we will examine
the existence of periodic and quasiperiodic (a special type of divergence) sequences
for certain bases.