CANT 2015 Abstracts
New York Number Theory Seminar
Thirteenth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 19-22, 2015

Paul Baginski, Fairfield University
Title: “Finding elements with prescribed factorization lengths”
Abstract: In algebraic number rings, elements can have nonunique factorization into
irreducible elements. For example, it is possible for an element x to have two distinct
factorizations as a product of five irreducibles. It is also possible for an element y
to factor as a product of two irreducibles, and separately, to factor as a productof three irreducibles. In the first case, the element x has only one factorization
length, 5, but it occurs twice, while in the second case, the factorization lengthsof y are {2, 3}. If one fixes the ring R, which finite subsets of N occur as sets
of factorization lengths of elements x ∈ R? Moreover, can one specify how often
the individual factorization lengths can occur? The answers to these questions ingeneral are quite difficult and depend heavily on combinatorial properties of theclass group of the ring.
One can ask these same questions in more general settings, such as Dedekind
domains and Krull monoids, which also possess class groups that reflect a richfactorization theory. Surprisingly, when the class group is infinite, it is far more
tractable to determine which length sets and multiplicities occur. We will givenatural examples from number theory and module theory of Krull monoids with
infinite class group, and then present the current state of affairs for finding elementswith prescribed factorization lengths and multiplicities. We will also describe theimplications of these results for the classic case of algebraic number rings, where
the class group is finite.

Bela Bajnok, Gettysburg College
Title: “Results and conjectures about sumsets in abelian groups”
Abstract: For a positive integer h and a subset A of a given finite abelian group,
we let hA, h± A, and hˆA denote the h-fold sumset, signed sumset, and restricted
sumset of A, respectively. In this talk we review what is known and not yet known
about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct andinverse problems.
1Dakota Blair, CUNY Graduate Center
Title: “Recurrence identities of b-ary partitions”
Abstract: Solving the b-ary partition problem, counting the number pb (n) of partitions of n into powers of b, is a pursuit which dates back to Euler. The function
pb (n) satisfies a recurrence, and this talk will examine a family of identities which
can be deduced by iterating the recurrence in a suitable way. These identities canthen be used to calculate pb (n) for large values of n. Further, these identities correspond to generating function identities involving a sequence of polynomials whichhave suggestive connections to Eulerian polynomials.

Lisa Bromberg, CUNY Graduate Center
Title: ”Navigating the Cayley graph of SL2 (Fp ) and applications to hashing”
Abstract: Hashing with matrices refers to a simple idea of using a pair of matrices, A
and B (over a finite ring), to hash the “0” and “1” bit, respectively, and then to hashan arbitrary bit string in the natural way, by using multiplication of matrices. Since
there are many known pairs of 2 X 2 matrices over Z that generate a free monoid,
this yields numerous pairs of matrices over Fp , for sufficiently large primes p, thatare candidates for collision-resistant hashing. However, this trick can “backfire”,
and lifting matrix entries to Z may facilitate finding a collision. This “lifting attack”
was successfully used by Tillich and Zemor in the special case where two matrices
A and B generate (as a monoid) the whole group SL2 (Z). However, in this paper
we show that the situation with other, “similar”, pairs of matrices from SL2 (Z) isdifferent, and the “lifting attack” can (in some cases) produce collisions in the group
generated by A and B, but not in the positive monoid. Therefore, we argue that forthese pairs of matrices, there are no known attacks at this time that would affectsecurity of the corresponding hash functions. We also give explicit lower bounds on
the length of collisions for hash functions corresponding to some particular pairs ofmatrices.

Mei-Chu Chang, University of California,
Title: “Multiplicative orders and distribution of points on varieties mod p”
Abstract: We will discuss mod p versions of Lang’s conjecture on torsion points onvarieties and related questions.

Scott Chapman, Sam Houston State University
Title: “On the catenary and tame degrees on a numerical monoid”
Abstract: Let M be a commutative cancellative monoid. For m a nonunit in
M , the catenary degree of m, denoted c(m), and the tame degree of m, denoted
t(m), are combinatorial constants that describe the relationships between differing
irreducible factorizations of m. These constants have been studied carefully inthe literature for various kinds of monoids, including Krull monoids and numericalmonoids. In this talk, we show for a given numerical monoid S that the sequences{c(s)}s∈S and {t(s)}s∈S are both eventually periodic. We show similar behavior
for several functions related to the catenary degree which have recently appeared inthe literature. These results nicely complement the known result that the sequence{∆(s)}s∈S of delta sets of S also satisfies a similar periodicity condition. We also
compute the catenary degree of elements contained in numerical monoids generated
by arithmetic sequences.

David John Covert, University of Missouri - St. Louis
Title: “Results on the Erdos-Falconer distance problem and its generalizations”
Abstract: We study the finite field and finite ring analogues of the Erdős-Falconer
distance problem. We state the first known results for general finite rings, andwe study a generalization of the distance problem, which we call the k-resultant
problem. Our results on k-resultant sets hint that the finite field distance problem
can be improved in even dimensions d ≥ 4. The work on the k-resultant set is joint
work with Doowon Koh and Youngjin Pi.

Robert Donley, Queensborough Community College (CUNY)
Title: “Toric varieties: Lectures 1 and 2”
Abstract: These are the first and second lectures in a series of four introductory
talks on toric varieties and their relation to convex polytopes. The last two lectures
will be given by Bart Van Steirteghem

Alex Gamburd, CUNY Graduate Center
Title: “Markoff triples and strong approximation”
Abstract. We study the connectedness of the set of solutions (mod p) of the Markoff
equation x2 + y 2 + z 2 = 3xyz under the action of the group of morphisms generated
by coordinate permutations and Vieta involutions. In particular, it is shown thatfor almost all primes the induced graph is connected. Similar results for composite
moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. Joint work with
J. Bourgain and P. Sarnak.

Leonid Gurvits, City College (CUNY)
Title: “On the complexity of the mixed volume of parallelograms”
Abstract: I will first review the only and inherently randomized poly-time algorithm
to approximate the mixed volume M V (K1 , ..., Kn ) of n convex bodies K1 , ..., Kn
in Rn within the relative error en . If the affine dimensions
aff(Ki ) ≤ 2, then the al√
Ki are parallelograms
gorithm approximates within the relative error (1 + 2)n . IfPthen their mixed volume is essentially the exponential sum S⊂{1,...,n} | det(AS )|,where A is some matrix and AS are its principal submatrices.
I will explain a few hardness and universality results that prohibit ”natural” approaches and will describe a deterministic poly-time algorithm to compute themixed volume of parallelograms within the relative error 2n .
Some open problems will be stated.

Sandie Han, New York City Tech (CUNY)
Title: “Orphans of the Calkin-Wilf trees for linear fractional transformations (LFT’s)”
Abstract: The determination of the orphan root of LFT’s on the infinite binary
Calkin-Wilf trees and tracing their paths were motivated by recent results of Nathanson
and joint work with Masuda, Singh, and Thiel.
 In particular, we will consider
a b
az+b
cz+d and its associated matrix T (z) = c d , where the elements are nonnegative integers acting on the set of complex numbers z ∈ C and the determinantdet (T (z)) 6= 0. We will show by considering the continued fraction representationsof a/b, b/d and T (z) how we can easily determine and trace the orphan roots of
LFT’s.

Charles Helou, Penn State - Brandywine
Title: “Characteristic, counting, and number of representations functions”
Abstract: Given a set A of natural numbers, i.e., nonnegative integers, there arethree distinctive functions attached to it, each of which completely determines A.
These are the characteristic function χA (n) which is equal to 1 or 0 according as thenatural number n lies or does not lie in A, the counting function A(n) which gives
the number of elements a of A satisfying a ≤ n, and the number of representations
function rA (n) which counts the ordered pairs (a, b) of elements a, b ∈ A such thata+b = n. We establish direct relations between these three functions. In particular,
we express each one of them in terms of each other one.

Alex Iosevich, University of Rochester
Title: “The Fuglede conjecture holds in Zp × Zp ”
Abstract: We prove that every function mapping E ⊂ Z2p , p prime, to C canbe expressed as a linear combination of characters orthogonal with respect to E
if and and only if E tiles Z2p by translation. The history of this problem in avariety of settings and some background material will also be discussed. Geometriccombinatorics, properties of the Fourier transform and elementary Galois theory
play an important role in the proof.

Renling Jin, College of Charleston
Title: “High density syndeticity”
Abstract: It was proven fifteen years ago that if two sets of natural numbers A andB have positive upper Banach density, then the sumset A + B must be piecewise
syndetic. Since then, many generalizations have been achieved. In this talk, wewill present a collection of new results concerning the high levels of syndeticity for
A + B if A and/or B has positive upper/lower asymptotic density.

Nathan Kaplan, Yale University
Title: “Jacobians of random graphs”
Abstract: The Jacobian of a graph G, sometimes called the Sandpile group or
critical group, is a finite abelian group given by the cokernel of the combinatorial
Laplacian of G. This group carries certain arithmetic information about the graph,for example, its order is the number of spanning trees of G. We will discuss the
distribution of Jacobians of Erdős-Renyi random graphs and will see connections to
Cohen-Lenstra heuristics for class groups of quadratic fields and to Hall-Littlewood
polynomials.

Mizan Khan, Eastern Connecticut State University
Title: “Some miscellaneous remarks about modular hyperbolas”
Abstract: Consider the point set
Hn = {(x, y) : xy ≡ 1(mod n), 1 ≤ x, y ≤ n − 1}.

We will use the symmetries of this set to give yet another proof that −1 is aquadratic residue modulo p if and only if p ≡ 1 (mod 4). We will then discuss theproof of the result that the number of vertices of the convex hull of Hn is nearly
always greater than 2(τ (n − 1) − 1).

Sandra Kingan, Brooklyn College (CUNY)
Title: “Growth rates and decomposition results in matroids”
Abstract: In this talk I will describe the close connection between the growth rate ofa minor-closed class of matroids and the decomposition of the class. The growth ratefunction of a minor-closed class of matroids is the maximum number of elements ina simple rank-r matroid in the class if the number is finite, and infinity otherwise. A
decomposition result for a minor-closed class describes the key 3-connected matroids
in the class, which are often infinite families of matroids or even smaller minor-closed
classes, and specifies how all the matroids are constructed from them. Having a
decomposition result for an excluded minor class leads to the growth rate functionof the class, but not necessarily the other way around. Specifically, I will discuss
my recent paper on the growth rate of a useful class of binary matroids and thetechniques used to decompose it.

Diego Marques, University of Brasilia
Title: “The solution of a problem posed by Kurt Mahler for real analytic functions”
Abstract: In 1902, P. Stackel proved the existence of a transcendental function f (z),analytic in a neighborhood of the origin, and with the property that both f (z) and
its inverse function assume, in this neighborhood, algebraic values at all algebraic
points. Based on this result, in 1976, K. Mahler proposed to investigate the existence of such functions which are analytic in C. In this work, we provide an answer
for this question in the real case by showing the existence of hypertranscendental
real analytic functions taking, together with its inverse, the set of all real algebraic
numbers into itself. Moreover, we can replace the set of algebraic numbers by any
countable and dense subset of R.

Ariane Masuda, New York City Tech (CUNY)
Title: “The (u, v)-Calkin-Wilf tree”
Abstract: The Calkin-Wilf tree is an infinite binary tree whose vertex set consistsof all positive rational numbers. In this talk we will present a generalization of the
Calkin-Wilf tree and show how some properties can be extended to the new setting.
This is based on joint work with Han, Singh, and Thiel.

Nathan McNew, Dartmouth College
Title: “The convex hull of the prime number graph”
Abstract: Let pn denote the n-th prime number, and consider the prime number
graph, the collection of points (n, pn ) in the plane. Pomerance uses the points lying
on the boundary of the convex hull of this graph to show that there are infinitely
many n such that p2n < pn−i + pn+i for all i < n. More recently, the primes on theboundary of this convex hull have been considered by Tutaj. We resolve several
conjectures of Pomerance and Tutaj, and relate these primes to other interesting
subsets of the prime numbers as well as other related forms of ’extreme’ primes.

Steven J. Miller, Williams College
Title: “From Fibonacci quilts to Benford’s law through Zeckendorf decompositions”
Abstract: Zeckendorf’s theorem states that every integer can be writen uniquely asa sum of non-adjacent Fibonacci numbers {Fn } (where F1 = 1 and F2 = 2); we callthis a legal decomposition. We report on some recent progress on generalizations
and related questions. In particular, we discuss two very different situations where
Benford’s law of digit bias emerges (which states that the probability of observing
a first digit of d is log10 (1 + 1/d)). The first is in the distribution of leading digitsof the summands in the Zeckendorf decompositions of integers; we concentrate on
the Fibonacci case, but the proof extends to other difference equations. The second
involves another sequence generated by a difference equation, which can be interpreted as the unique sequence arising from imposing a rule for a legal decomposition
from the geometry of the Fibonacci spiral. In this situation we lose uniqueness of
decomposition. We prove that approximately 92.6algorithm terminates in a legal
decomposition here, and the average number of legal decompositions of numbers at
most Fn follows Benford’s law.

Mel Nathanson, Lehman College (CUNY)
Title: “Asymptotic approximate groups”
Abstract: Let A be a finite subset of an abelian group. There exists h0 = h0 (A)such that hA is an approximate group for all h ≥ h0 .

Kevin O’Bryant, College of Staten Island (CUNY)
Title: “Ordinals arising from n-alpha sequences”
Abstract: An ordinal set is a set of reals that does not have an infinite decreasing
subsequence. Given a set A of natural numbers and a real number x, the set
x ∗ A := {ax mod 1 : a ∈ A} may or may not be an ordinal set. We investigate
the connections between the set A, the multiplier x, and the order type of x ∗ A.
Joint work with Dakota Blair and Joel Hamkins.

Cormac O’Sullivan, Bronx Community College, CUNY
Title: “Rademacher’s conjecture and Sylvester’s waves”
Abstract: Rademacher conjectured that the coefficients in the partial fraction decomposition of the generating function for the restricted partition function pN (n),(counting partitions of n into at most N parts), should converge as N goes to infinity to the corresponding coefficients for the unrestricted partition function p(n). We
describe the recent disproof of this conjecture and also current work showing thatthe method may be used to quantify how well the first Sylvester waves approximate
pN (n) and p(n).

Jasmine Powell, Northwestern University, andMadeleine Weinstein, Harvey Mudd College
Title: “Geometric-progression-free sets over quadratic number fields”
Abstract: A problem of recent interest has been to study how large subsets of thenatural numbers can be while avoiding 3-term geometric progressions. Building on
recent progress on this problem, in this talk we consider the analogous problem
over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometricprogressions, generalizing a greedy construction used by Rankin. When uniquefactorization fails or over a real quadratic number field, we instead look at subsets
of ideals of the ring of integers and describe the densities of these sets in terms ofvalues of the Dedekind zeta function. Next, we consider geometric-progression-freesets with large upper density and obtain upper and lower bounds for the upper
density of geometric-progression-free subsets. This is joint work with Andrew Best,
Karen Huan, Steven J. Miller, Nathan McNew, and Kimsy Tor.

Alex Rice, University of Rochester
Title: “Difference sets and polynomials”
Abstract: In a series of papers in the 1970s, Sárközy proved that any set of integersof positive upper density necessarily contains two distinct elements which differ by aperfect square, as well as two elements which differ by one less than a prime number,confirming conjectures of Lovász and Erdős, respectively. In this talk, we provide
a brief survey of the extensive literature that has developed on improvements andextensions of these results, culminating in a brand new ”super theorem” which
expands to sums of polynomials, improves certain quantitative bounds, and includes
most previous results as special cases. This is joint work with Neil Lyall.

Steven Senger, Missouri State University
Title: “Upper bounds on pairs of dot products in various vector spaces”
Abstract: Given a subset of points in either R2 or a vector space over a finite
field, we give bounds on how many triples of points determine a fixed pair of dot
products.

Satyanand Singh, New York City Tech (CUNY)
Title: “An accidental sequence”
Abstract: We will study a Diophantine Equation raised by Bennett which is pivotal
in establishing that the perfect powers of five have few ternary digits in their ternary
expansions. In particular we will consider the equation 3a + 3b + 2 = n5 , where thegcd(n, 3) = 1 and a > b > 0 and establish its insolubility when the pair (a, b) has
opposite parity and even parity by elementary methods. In the case for the pair(a, b) with odd parity there is one known solution a = 3, b = 1 and n = 2. We will
illustrate by computation that no other solutions exist for n < 2 + 6 · 106 and touchupon the birth of the Sequence A224920. We will then discuss pertinent related
open problems.

Jonathan Sondow, New York
Title: “Irrationality and transcendence of alternating series via continued fractions”
Abstract: We give conditions for irrationality of the sum of an alternating series,
and introduce a “simple” condition for its transcendence. The proofs use continued
fractions, irrationality measure, and the Thue-Siegel-Roth theorem on rational approximations to algebraic numbers. The simple continued fractions for an infinite
family of naturally-occurring transcendental numbers are given explicitly. We also
prove irrationality and transcendence for families of non-alternating series, using
partial sums instead of continued fractions. Mentioned along the way are Fermat,
Fibonacci, and Liouville numbers, π, e, primorials, Sylvester’s sequence, and someconjectures.

Johann Thiel, New York City Tech (CUNY)
Title: “Orphans in generalized Calkin-Wilf trees”
Abstract: In this talk we will consider two generalizations of the Calkin-Wilf tree.
In particular, we will discuss some new results on the existence and enumeration oforphan roots. This is based on joint work with Han, Masuda, and Singh.

Yuri Tschinkel, NYU
Title: ”Height zeta functions”
Abstract: I will explain some geometric and analytic techniques applied to counting
problems in the theory of diophantine equations in many variables.

Bart Van Steirteghem, Medgar Evers College (CUNY)
Title: “Toric varieties: Lectures 3 and 4”
Abstract: These are the third and fourth lectures in a series of four introductory
talks on toric varieties and their relation to convex polytopes. The first two lectures
will be given by Robert Donley.