CANT 2017 Abstracts
Fifteenth Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 23–26, 2017
Ali Armandnejad, Vali-e-Asr University of Rafsanjan, IranTitle: Number of signature matrices on each component of J-orthogonal matrices
Abstract: Let Mn be the set of all n × n real matrices. A matrix J ∈ Mn is said
to be a signature matrix if J is diagonal and its diagonal entries are ±1. A matrix
A ∈ Mn is said to be a J-orthogonal matrix if there exists a signature matrix J
such that A> JA = J. It is a well known topological result that for J 6= ±I, the setof all n × n J-orthogonal matrices has four connected components. In this note wefind the number of signature matrices that is placed on each component of the setof n × n J-orthogonal matrices.
Abdul Basit, Rutgers - New Brunswick
Title: Recent improvements in sum-product estimates
Abstract: The sum-product conjecture of Erdős-Szemerédi states that for any finite
subset A of the reals max{|A + A|, |AA|} ≥ |A|2− . Here A + A (resp. AA) is theset of all pairwise sums (resp. products) of elements of A. We will discuss somerecent improvements for various sum-product type estimates.
Sam Chow, University of York, England
Title: Bohr sets and multiplicative diophantine approximation
Abstract. In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an
inhomogeneous fibre version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani.
The idea is to find large generalised arithmetic progressions within inhomogeneous
Bohr sets, extending a construction given by Tao. This precise structure enables us
to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand,
via the geometry of numbers.
David Chudnovsky and Gregory Chudnovsky, NYU
Title: Near isomorphism of graphs1Colin Defant, University of Florida
Title: A new proof of Euler’s formula for ζ(2k)Abstract: Let fk : R → R be the function that is periodic with period 2π andsatisfies fk (x) = xk whenever −π < x ≤ π. Analyzing the Fourier coefficients ofthese functions yields a quick inductive proof that ζ(2k) is a rational multiple ofπ 2k for every positive integer k. By attending to details, we provide a new inductive
proof of Euler’s classical formula(−1)k+1 B2k (2π)2k.
2(2k)!
Along the way, we also find a new identity involving Bernoulli numbers.
This is joint work with K. Alladi.
ζ(2k) =

Robert Donley, Queensborough Community College (CUNY)
Title: Clebsch-Gordan coefficients in characteristic zero
Abstract: Clebsch-Gordan coefficients for SL(2, C) play a central role in representation theory, classical particle physics, and special functions. Known closed formulas(Wigner, Racah, Shapiro) involve an alternating sum, factorials, and square roots.
By adjusting the usual conventions, we obtain a closed formula with rational values,
an associated generating function, and an efficient algorithm for computation by
hand.
Joseph Gunther, CUNY Graduate Center
Title: Slicing the stars: Counting algebraic numbers by degree and height”
Abstract: Masser and Vaaler have given an asymptotic formula for the numberof algebraic numbers of given degree d and increasing height. This problem wassolved by counting lattice points (which correspond to minimal polynomials overZ) in a homogeneously expanding star body in Rd+1 . The volume of this star
body was computed by Chern and Vaaler, who also computed the volume of thecodimension-one “slice” corresponding to monic polynomials – this led to results ofBarroero on counting algebraic integers. We’ll discuss how to estimate the volumeof higher-codimension slices, which allows us to count units, algebraic integers ofgiven norm, trace, norm and trace, and more.
This is joint work with Robert Grizzard.
Brandon Hanson, Pennsylvania State University
Title: Integers not represented by diagonal forms of large degree
Abstract: Waring’s Problem tells us that for fixed k and large enough s, every
number can be written can be written as a sum of s perfect kth powers. If, instead,
we fix s and let k grow, such a result is blatantly false - to represent N , we arelimited to the kth powers bounded by N 1/k , and there just aren’t enough kth
powers to go around. On the other hand, if we work with sums and differences,
this fact is no longer present (we are now representing numbers with an indefiniteform). None the less, we still expect that the number of integers represented to bequite small when k is large relative to s. Such a statement is only known to be true
conditional on the generalized abc-conjecture. In joint work with Asif Zaman, we
aim for a more modest, though unconditional, result: The numbers represented bya diagonal form of large degree are usually not too dense.

Charles Helou, Pennsylvania State University - Brandywine
Title: Cyclotomic expressions for representation functions
Abstract: Given a subset A of the set N = {0, 1, 2, · · · } of natural numbers or of theresidue class ring Z/N Z, where N is a fixed positive integer, we introduce certain
exponential sums associated with A. They allow us to obtain expressions for theclassical functions that characterize A, in particular the additive representation
function of A. The expressions are first given directly in terms of the exponential
sums, but then also in terms of algebraic forms such as the trace forms in subfields
of a cyclotomic field. These sums are also of intrinsic interest, and we note some of
their properties.
Brian Hopkins, St. Peter’s University
Title: Magic configurations
Abstract: An (nk ) configuration is a set of n points and n lines such that every
point lies on k lines and every line contains k points. These geometric/combinatorial
objects date back to Pappus and saw a flurry of attention in the late 19th century
but then suffered a lull until a 1990 renaissance due to Branko Grünbaum. Here we
consider a question motivated by magic squares and the like: Can the points of aconfiguration be labeled with distinct integers such that the sums for each line are
the same? We restrict our attention to k = 3. Results include two approaches tofinding the minimal n for which a magic (n3 ) configuration exists and a construction
showing that there are infinitely many such configurations.
This is joint work with Michael Raney of Georgetown University.
Robert Hough, SUNY at Stony Brook
Title: The shape of cubic fields
Abstract: The ring of integers of a cubic field K is a lattice in the canonicalembedding of K in R3 or R × C. I will discuss ongoing work which studies thisshape spectrally when cubic forms are ordered by increasing size of the discriminant.
The method builds on the ζ-function approach of Shintani and Taniguchi-Thorne.
Alex Iosevich, University of Rochester
Title: On a two-parameter variant of the Erdős distance problem
Abstract: Given E ⊂ R2 , let B2,2 (E) = {(|x0 − y 0 |, |x00 − y 00 |) : x = (x0 , x00 ), y =(y 0 , y 00 ) ∈ E}. The question we ask is, if #E = n, how small can #B2,2 (E) possiblybe?
William J. Keith, Michigan Technological University
Title: Major index over descent for pattern avoidance classes: unimodality andother combinatorial features
Abstract: The Eulerian-Mahonian bivariate generating function for the joint distribution of the major index and descent statistics of permutations of length n isa classical object with many well-established features. Less studied but perhaps
equally interesting is the same generating function with the summation restricted
to pattern avoidance classes. We shall discuss the usual questions of symmetricity,
unimodality, and recurrences, as well as some observations of interesting combinatorial features to motivate future research.

Mizan R. Khan, Eastern Connecticut State University
Title: An amuse-bouche arising from a result of Mordell
Abstract: Let p be a prime with p ≡ 3 (mod 4), and consider the modular hyperbola
Hp = {(x, y) : xy ≡ 1 (mod p), 1 ≤ x, y ≤ p − 1}.
We have thatp−1
! ≡ (−1)a (mod p).
2Mordell, in the February 1961 issue of the Amer. Math. Monthly, gave an elegant
criterion to determine the parity of a. We apply this criterion to determine the parity of elements of Hp that lie in the triangle with vertices (0, 0), (0, p/2), (p/2, p/2).
Byungchan Kim, SeoulTech, Republic of Korea
Title: (q, t)-Delannoy numbers and integer partitions
Abstract: We introduce an overpartition analogue of Gaussian polynomials m+nnq,t
as the generating function for overpartitions fitting inside an m×n rectangle, whichis also a (q, t)-analogue of Delannoy numbers. First we obtain finite versions of classical q-series identities such as the q-binomial theorem and the Lebesgue identity, as
well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite
theta function and prove the (q, t)-log concavity of m+n. We particularly emn
q,tphasize the role of combinatorial proofs and the consequences of our results on
Delannoy numbers.
This is joint work with Jehanne Dousse.
Hershy Kisilevsky, Concordia University, Canada
Title: Sign matrices and quadratic residue matrices
Abstract: Sign matrices are n × n matrices with diagonal entries equal to zero andoff diagonal entries ±1. A sign matrix is a Quadratic Residue (QR) matrix if theentries are Legendre Symbols. QR matrices arise in the study of the decompositionof the ramified primes in tamely minimally ramified extensions K/Q which arecomposites of quadratic extensions. We characterize QR matrices among the n × nsign matrices and show that the proportion of QR matrices becomes vanishingly
small as n → ∞.
This is joint work with D.S. Dummit and E. Dummit.
Sándor Kiss, Budapest University of Technology and Economics, Hungary
Title: On the structure of sets which has coincide representation functions
Abstract: For a set of nonnegative integers S let RS (n) denote the number ofunordered representations of the integer n as the sum of two different terms from
S. Together with Csaba Sándor we partially described the structure of the sets,
which has coincide representation functions by using Hilbert cubes.

Nana Li, Bard College at Simon’s Rock
Title: On the union-closed sets conjecture
Abstract: A family of finite sets is called union-closed if it contains the union of any
two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states thateach union-closed family contains an element that belongs to at least half of themembers of the family. In this paper, we will talk about the recent development of
this conjecture and several other related conjectures.
Jared Lichtman, Dartmouth College
Title: Lying on the Fermat primality test
Abstract: We investigate the probability that a random odd composite number
passes a random Fermat primality test, improving on earlier estimates in moderate
ranges. For example, with random numbers to 22 00, our results improve on priorestimates by close to 3 orders of magnitude. We also provide the true probabilities insmaller ranges. These results, combined with random sampling estimates in larger
ranges, suggest that the true probabilities are considerably smaller than current
upper bounds.
This is joint work with Carl Pomerance.
Neil Lyall, University of Georgia
Title: Embedding distance graphs into sets of positive density, I
Abstract: A distance graph is a graph G = (V, E) whose vertex set V = {v1 , ..., vn }is contained in a Euclidean space. We say that G can be embedded in a set A if
A contains a set W = {w1 , ..., wn } such that |wi − wj | = |vi − vj | for all edges(i, j) ∈ E. We show that all large dilates of G can be embedded into any set A of
positive upper density of the d-dimensional Euclidean space as long as deg(G) < d.
This latter condition means for any j the number of edges (i, j) for which i < jisless than d. We show that similar results are possible for subsets of the integer
lattice in dimensions d > C deg(G)2 .
Akos Magyar, University of Georgia
Title: Embedding distance graphs into sets of positive density, II
Abstract: A distance graph is a graph G = (V, E) whose vertex set V = {v1 , ..., vn }is contained in a Euclidean space. We say that G can be embedded in a set A if
A contains a set W = {w1 , ..., wn } such that |wi − wj | = |vi − vj | for all edges(i, j) ∈ E. We show that all large dilates of G can be embedded into any set A of
positive upper density of the d-dimensional Euclidean space as long as deg(G) < d.
This latter condition means for any j the number of edges (i, j) for which i < jisless than d. We show that similar results are possible for subsets of the integer
lattice in dimensions d > C deg(G)2 .

Michael Maltenfort, Northwestern University
Title: Characterizing additive systems
Abstract: An additive system is a collection of sets that gives a unique way torepresent either all nonnegative integers, or all nonnegative integers up to somemaximum. A structure theorem of de Bruijn gives a certain form for an additive
system of infinite size. This form is not, in general, unique. We improve de Bruijn’stheorem by finding a unique form for an additive system of arbitrary size. Our proof
gives a concrete construction that allows us to test easily whether a collection ofsets is an additive system. We also show how to determine most of the structureof an additive system if we are only given its union.
Azita Mayeli, Queensborough Community College (CUNY)
Title: Recent developments in Gabor orthogonal bases on finite vector spaces
Abstract: We study orthogonal Gabor bases in Zdp , the d-dimensional vector spaceover the cyclic group of prime order. These are bases of the form {g(x − a)χ(x · b)},a ∈ A, b ∈ B. The function g is called the window, A a set of translations and B
a set of modulations. The investigation has analytic, number theoretic and combinatorial aspects. The analytic arguments comes in the form of Fourier analytic
inequalities, the number theoretic aspects involve the analysis of exponential sums
and the combinatorial arguments involve tiling and the study of directions sets.
This is a joint work with Alex Iosevich and Jonathan Pakianathan.
Nathan McNew, Towson State University
Title: Primitive and geometric-progression-free sets without large gaps
Abstract: We show how the probabilistic method can be used to construct primitive
sets (sets of integers where no integer divides another) with relatively small gaps
between consecutive terms, substantially smaller than is known to hold for theprimes. We then show how the same techniques can be used to improve the bounds
obtained by He for geometric-progression-free sets.
Steven J. Miller, Williams College
Title: Ramsey theory over number fields and quaternions
Abstract: In Ramsey theory one wishes to know how large a collection of objects
can be while avoiding a particular substructure. A problem of recent interest has
been to study how large subsets of the natural numbers can be while avoiding
3-term geometric progressions. Building on recent progress on this problem, weconsider the analogous problem over quadratic number fields. We first constructhigh-density subsets of the algebraic integers of an imaginary quadratic number
field that avoid 3-term geometric progressions. When unique factorization fails orover a real quadratic number field, we instead look at subsets of ideals of the ringof integers. Our approach is to construct the sets greedily, a generalization of thegreedy set of rational integers considered by Rankin. We then describe the densitiesof these sets in terms of values of the Dedekind zeta function. If time permits wewill discuss similar results in other settings (such as the quaternions).
This work is from REU projects supervised jointly with Nathan McNew.

Mel Nathanson, CUNY
Title: Problems in additive number theory and applications of commutative algebra
Abstract: This will be a survey of open problems in additive number theory, with
examples of the application of classical theorems in commutative algebra to obtain
additive results.
Hans Parshall, University of Georgia
Title: Spherical quadrilaterals over finite fields
Abstract: Graham conjectures that a finite Euclidean point set is Ramsey if andonly if it is spherical, but we do not even know if this is true for a generic four point
subset of a circle. We will discuss analogous problems in vector spaces over finite
fields. In joint work with Neil Lyall and Ákos Magyar, we show that counting operators associated to spherical configurations are controlled by geometric uniformity
norms. This approach allows us to establish a finite field analogue of the first open
case of Graham’s conjecture.
Giorgis Petridis, University of Georgia
Title: An inverse sum-product result in arbitrary fields
Abstract: One of the fundamental results in sum-product theory is, given two finite
sets A, X in a field F, to obtain a non-trivial lower bound for the cardinality of theset
A + XA = {a + xb : a, b ∈ A, x ∈ X}.
The lower bound |A||X|1/2 is, for example, known when F = R or F = Fp , aprime order finite field (mild conditions on |X|, |A| are necessary).
The presence of subfields causes problems in positive characteristic. Take forexample X to be a subset of a proper (and finite) subfield and A to be a vector
space over the subfield. Then A + XA = A + A = A. Similarly, if A is a subset ofthe vector space with relative density 1/K, then |A + XA| ≤ K|A|.
In joint work with Brendan Murphy we established an inverse result by showingthat these are the only examples where |A + XA| can be nearly minimum. Morespecifically, we proved that if |A + XA| ≤ K|A|, then X is a subset of a (finite)
proper subfield and A has “large” relative density in the vector space over thesubfield it generates. Large here means a small integral power of K −1 , which is
close to being optimal.
We will discuss the relation to Bourgain’s powerful bounds on the number ofsolutions to the equation
a1 − a2 = x(a3 − a4 ) ai ∈ A, x ∈ X
and on products of differences in arbitrary finite fields.
Mengquing Qin, Missouri State University
Title: Bounds on AA + AA in terms of AA in finite fields
Abstract: We present some results relating the relative sizes of AA and AA +AA, where A is a subset of a finite field. These follow as consequences of results
due to Hart, Iosevich, and Solymosi. We also provide some simple corollaries andapplications.

Sinai Robins,
Title: Discrete volumes and 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 angle
sum of a polytope, defined as follows. Place a very small sphere, centered at eachlattice point of a given lattice L ⊆ Rd and consider the proportion of that spherethat intersects the polytope P, called a local solid angle. Now translate this small
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 dilations t, for
any real polytope (so that its vertices do not necessarily lie on the lattice L). One of
our ingredients is a detailed description of the Fourier transform of the polytope P.
This transform method uses the formula of Stokes, which is a way of integrating by
parts in several variables. Another key tool for us is the Poisson summation formula,applied to smoothings of the indicator function of the polytope P. It turns out thatthe combinatorics of the face poset of P plays a central role in the description of
the Fourier transform of P, and in keeping track of the infinite series that pop outof Poisson summation. We also obtain a surprisingly simple closed form for thecodimension-1 coefficient of the solid angle polynomial AP (t), for any real dilate ofa rational polytope, extending previously known results about this codimension-1
coefficient. This is joint work with Ricardo Diaz and Quang-Nhat Le.
Ryan Ronan, CUNY Graduate Center
Title: An asymptotic for the growth of Markoff-Hurwitz tuples satisfying a congruence relation
Abstract: Consider the Markoff-Hurwitz equation
x21 + x22 + · · · + x2n = ax1 x2 · · · xn
for integer parameters n ≥ 3, a ≥ 1. In this talk, we establish an asymptotic countfor the number of integral solutions with max{x1 , x2 , . . . , xn } ≤ R and lying in agiven congruence class modulo a square-free number q for which a particular transitivity property holds. When n = 3, a = 1 (or, equivalently, a = 3) this equation is
known simply as the Markoff equation, and a number which appears as a solution
to the Markoff equation is referred to as a Markoff number. Bourgain, Gamburd,
and Sarnak recently showed that almost all Markoff numbers are composite. A
crucial ingredient in their proof is a result due to Mirzakhani which provides anasymptotic formula for the number of Markoff triples below a given height and lying
in a given congruence class. We use methods from symbolic dynamics to obtain theanalogous count in the general case of arbitrary n and a.
This is joint work with Alex Gamburd and Michael Magee.

Csaba Sándor, Budapest University of Technology and Economics, Hungary
Title: A lower bound of Ruzsa’s number related to the Erdős-Turán conjecture
Abstract: For a set A ⊆ N and n ∈ N, let RA (n) denote the number of orderedpairs (a, a0 ) ∈ A × A such that a + a0 = n. The celebrated Erdős-Turán conjecturesays that, if RA (n) ≥ 1 for all sufficiently large integers n, then the representation
function RA (n) cannot be bounded. For any positive integer m, Ruzsa’s number
Rm is defined to be the least positive integer r such that there exists a set A ⊆ Zm
with 1 ≤ RA (n) ≤ r for all n ∈ Zm . In 2008, Chen proved that Rm ≤ 288 for all
positive integers m. Together with Quan-Hui Yang we proved that Rm ≥ 6 for all
integers m ≥ 36. This lower bound is nearly the best possible. We discuss related
problems as well.
James Sellers, Pennsylvania State University
Title: An infinite family of congruences for `-regular overpartitions
Abstract: We consider new properties of the combinatorial objects known as overpartitions (which are natural generalizations of integer partitions). In particular,
we establish an infinite set of Ramanujan-type congruences for the restricted overpartitions known as `-regular overpartitions. This significantly extends the recent
work of Shen which focused solely on 3regular overpartitions and 4regular overpartitions.
This is joint work with Abdulaziz M. Alanazi and Augustine O. Munagi of University of the Witwatersrand, South Africa.
Steve Senger, Missouri State University
Title: Rainbow triangles
Abstract: We prove that, under some mild conditions, any coloring of two-dimensional
vector space over a finite field satisfying some maximum color class constraints musthave a unit equilateral triangle with each vertex from a different color class. We
also discuss some related results.
Satyanand Singh, New York City Tech (CUNY)
Title: A sinusoidal twist with exponential influences
Abstract: The under damped oscillating spring mass dynamical system with displacement shaped like u(t) = P e−γt/2m cos(µt − δ) and no external force is mostinteresting as a discrete mathematical model with applications in quantum dynamics and cryptography. If we let m be the mass, γ the damping constant, c the spring
constant and t = τ1 > 0 the instant where the first extreme value occurs, we willshow that the distance S traveled by a particle modeled in this system results inthe unexpectedly nice closed form expression given below.
hi
S = P | cos(φ)| (1 + coth (γπ/4mµ)) e−γτ1 /2m − |cos(δ)| sgn (u(0)u(τ1 )) .

Jonathan Sondow, New York
Title: Power-sum denominators
Abstract: The power sum 1n + 2n + · · · + xn has been of interest to mathematicianssince classical times. Johann Faulhaber, Jacob Bernoulli, and others who followed
expressed power sums as polynomials in x of degree n + 1 with rational coefficients.
Here we consider the denominators of these polynomials, and prove some of theirproperties. A remarkable one is that such a denominator equals n + 1 times thesquarefree product of certain primes p obeying the condition that the sum of thebase-p digits of n + 1 is at least p. As an application, we derive a squarefree productformula for the denominators of the Bernoulli polynomials.
This is joint work with Bernd C. Kellner. Our paper will appear in the American
Mathematical Monthly in 2017/18.
Jack Sonn, Technion, Israel
Title: Quadratic residues and difference sets
Abstract: It has been conjectured by Sárközy that with finitely many exceptions,the set of quadratic residues modulo a prime p cannot be represented as a sumset{a + b : a ∈ A, b ∈ B} with non-singleton A, B ⊆ Fp . The case A = B of thisconjecture has been recently established by Shkredov. The analogous problem fordifferences remains open: is it true that for all sufficiently large primes p, the set ofquadratic residues modulo p is not of the form {a0 − a00 : a0 , a00 ∈ A, a0 6= a00 } withA ⊆ Fp ?
We attack here a presumably more tractable variant of this problem, which is
to show that there is no A ⊆ Fp such that every quadratic residue has a uniquerepresentation as a0 − a00 with a0 , a00 ∈ A, and no non-residue is represented in thisform. We have produced a number of necessary conditions for the existence ofsuch A, involving for the most part the behavior of primes dividing p − 1. These
conditions enable us to rule out all primes p in the range 13 < p < 1020 (theprimes p = 5 and p = 13 being conjecturally the only exceptions). The talk will
focus on one of these conditions, which emerges from an algebraic number theoretic
approach to the problem.
This is joint work with Vsevolod Lev.
Yoni Stancescu, Afeka College, Israel
Title: Selected problems and results in inverse additive number theory
Abstract: We shaill discuss a number of problems and results in combinatorial
number theory and we will mostly focus on multi-dimensional sets in torsion freegroups.
Stefan Steinerberger, Yale University
Title: Strange new interactions between analysis and number theory
Abstract: I will discuss three surprising interactions between analysis and number
theory: (1) a result about maximal, local averages of functions that seems to needthe Lindemann-Weierstrass theorem, (2) a curious sum of sines that knows thedifference between rational and irrational numbers, and (3) mysterious patterns inan old 1962 integer sequence of Stanislaw Ulam ($300 prize for an explanation).

Salvatore Tringali, Institute for Mathematics and Scientific Computing, University of Graz, Austria
Title: On the Interplay between arithmetic combinatorics and factorization theory
Abstract: From a classical point of view, factorization theory is all about various
phenomena arising from the non-uniqueness of factorization in atomic monoids (andrings), and their classification by an assortment of algebraic, arithmetic, or combinatorial invariants. The subject has become more and more popular since thepublication of Geroldinger and Halter-Koch’s 2006 monograph, which is entirely
devoted to the commutative and cancellative case.
The main goal of the talk is to provide some evidence in support of the ideathat arithmetic combinatorics and factorization theory can draw great benefit fromthe interaction with each other. More specifically, we will present an overview ofa couple of recent developments in this direction, which have also resulted into a
generalization of fundamental aspects of factorization theory to non-cancellative or
non-commutative settings:(1) Power monoids and restricted power monoids, whose arithmetic is, in somerelevant cases (namely, for Dedekind-finite, aperiodic monoids), naturally
linked to various properties of non-empty finite subsets of N that can or
cannot be written as a sumset of a prescribed number of irreducible (orprimitive) sets;(2) The structure theorem for directed families of sets of non-negative integers,
along with applications to the unions of sets of lengths of unit-cancellative
atomic monoids.
This is joint work with Fan (arXiv:1701.09152) and Fan, Geroldinger, and Kainrath(arxiv:1612.03116).
Yuri Tschinkel, Courant Institute, NYU
Title: Rationality problems
Abstract: I will discuss recent advances in algebraic geometry, inspired by number
theory. This is joint work with B. Hassett and A. Pirutka.
Ajmain Yamin, Bronx High School of Science
Title: “An n-dimensional Calkin-Wilf tree and its graph structure”
Abstract: The Calkin-Wilf tree is an infinite binary tree in which the root is labeled
1/1 and the children of an arbitrary node a/b are a/(a + b), (a + b)/b. A positiven-dimensional fraction is a formal symbol a1 /a2 /.../an /b, where ai , b ∈ Z+ and is in
reduced form if and only if gcd(a1 , a2 , ..., an , b) = 1. The n-dimensional Calkin-Wilftree is a generalization of the Calkin-Wilf tree, where the Calkin-Wilf tree is the
1-dimensional case. The n-CW tree has root 1/1/ · · · /1/1 and the children of anarbitrary node a1 /a2 /.../an /b are all the n-dimensional fractions x1 /x2 / · · · /xn /ywhich satisfy (a1 , a2 , · · · , an , b) replaces (x1 , x2 , · · · , xn , y) under one iteration of
the Euclidean Algorithm. It follows immediately that each reduced positive ndimensional fraction labels exactly once. We study the nontrivial graph structurethat emerges from this definition. A notion of associated partition is crucial. First,
we obtain an expression for the out-degree of an arbitrary node which only depends
on the associated partition. Next, we make a statement about which associatedpartitions are allowed in the child-set. Finally, we construct a finite directed graph,Diagram(n + 1), of associated partitions which gives the complete graph structureof the n-dimensional Calkin-Wilf tree.
Yifan Zhang, Central Michigan University
Title: A combinatorial proof for the generating function of the product of secondorder recurrence sequences
Abstract: In this paper, we establish a combinatorial method to obtain the generating function for the product of finitely many second-order recurrence sequences.
We give several examples including the generating function for the power of binomial coefficients and for the product of Fibonacci and Lucas numbers.
This is joint work with George Grossman.