CANT 2013 Abstracts
New York Number Theory Seminar
Eleventh Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate Center, Room C198
May 21-24, 2013

Abstracts of lectures

Paul Baginski, Smith College“Length multiset-complete Krull monoids”
Abstract: Let H be a Krull monoid or Krull domain, let G be its divisor classgroup, and let G0 ⊂ G be the classes containing prime divisors. It is well known
that each nonunit x ∈ H has only finitely many factorizations into irreducibles. If
x = a1 · · · an is a factorization z of x into irreducibles, the length of this factorization
is n = |z|. We elaborate upon the well-studied set L(x) of factorization lengths ofx to account for the number of factorizations of a given length. If Z(x) is the setof factorizations of x (a subset of the free monoid over the irreducibles of H), thenthe length multiset of x, denote LM(x), is the multiset {{ |z| : z ∈ Z(x) }}.
Kainrath has shown that if the Krull monoid H has infinite class group G and
G0 = G, then for any finite multiset S on N\{1}, there is an x ∈ H with LM(x) =S. Kainrath’s proof was nonconstructive. In this talk we will give the backgroundon Kainrath’s result and illustrate a constructive proof for G = Z. We will also
discuss recent work to extending Kainrath’s result to Krull monoids with G = Z
but G0 a proper subset of Z.

Arnab Bhattacharyya, DIMACS, Rutgers University“Every locally characterized affine-invariant property is testable”
Abstract: Let F = Fp for any fixed prime p ≥ 2. An affine-invariant property is aproperty of functions on F n that is closed under taking affine transformations of thedomain. We prove that all affine-invariant property having local characterizations
are testable. In fact, we show a proximity-oblivious test for any such property
P , meaning that there is a test that, given an input function f, makes a constantnumber of queries to f, always accepts if f satisfies P, and rejects with positive
probability if the distance between f and P is nonzero. More generally, we show thatany affine-invariant property that is closed under taking restrictions to subspaces
and has bounded complexity is testable.
We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized andis, hence, testable. For example, whether a function is a product of two degree-d1polynomials, whether a function splits into a product of d linear polynomials, andwhether a function has low rank are all examples of degree-structural properties
and are therefore locally characterized.
Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low
characteristic fields that decomposes any polynomial with large Gowers norm intoa function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of boththe testability results and the local characterization of degree-structural properties.
Joint work with Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar
Lovett.

Gautami Bhowmik, Université de Lille, France“Lattice polyhedra in additive combinatorics”
Abstract : One of the topics of additive combinatorics is the study of sequences andsets embedded in an ambient group and their behavior under the group operation.
In particular we are interested in cases where these sequences contain elements
which add up to zero, the group being written additively. I will mention someexisting algebraic and analytic methods in use and treat with some detail a new
method of using convex polyhedra for treating Fdp for small dimensions d and large
primes p.

Thomas Bloom, University of Bristol, UK“Arithmetic structure in polynomial rings”, Abstract: We apply the techniques
used recently with great success in Freiman’s theorem in the integers to study asimilar problem in Fq [t], the ring of polynomials over a finite field. The problem isthat given a finite set A for which A + t · A is ’small’, how much does A resemble an
Fq [t] progression, that is, an affine image of the set of polynomials with degree at
most N . We show that by adapting techniques used in the integer case we are able
to obtain near-optimal bounds in one respect, and discuss applications to problems
in arithmetic combinatorics over finite field vector spaces Fnp .

Tomas Boothby, Simon Fraser University, Canada“Almost critical products sets and small edge cuts in almost symmetric graphs”
Abstract: In a group G, two finite subsets A and B are ’critical’ if |AB| < |A| + |B|,and ’almost critical’ if |AB| = |A| + |B|. A recent paper of DeVos classifies criticalpairs (A, B). One type of critical pair corresponds to edge cuts S of size |S| <2d in vertex- and edge-transitive (almost symmetric) d-regular graphs. Towards
classifying almost critical pairs, we classify all corresponding edge cuts with |S| =2d. In this talk, I will discuss the correspondence between these pairs and edge cutsand present their classification.

Amanda Bower, University of Michigan-Dearborn, andVictor Luo, Williams College“Coordinate sum and difference sets of d-dimensional modular hyperbolas”
Abstract: Many problems in additive number theory, such as Fermat’s last theorem
and the twin prime conjecture, can be understood by examining sums or differencesof a set with itself. A finite set A ⊂ Z is considered sum-dominant if |A + A| >|A − A|. If we consider all subsets of {0, 1, . . . , n − 1}, as n → ∞ it is natural
to expect that almost all subsets should be difference-dominant, as addition iscommutative but subtraction is not; however, Martin and O’Bryant in 2007 provedthat a positive percentage are sum-dominant as n → ∞.
This motivates the study of “coordinate sum dominance”. Given V ⊂ (Z/nZ)2 ,we call S := {x+y : (x, y) ∈ V } a coordinate sumset and D := {x−y : (x, y) ∈ V } acoordinate difference set, and we say V is coordinate sum dominant if |S| > |D|. An
arithmetically interesting choice of V is H̄2 (a; n), which is the reduction modulo
n of the modular hyperbola H2 (a; n) := {(x, y) : xy ≡ a mod n, 1 ≤ x, y < n}.
In 2009, Eichhorn, Khan, Stein, and Yankov determined the sizes of S and D
for V = H̄2 (1; n) and investigated conditions for coordinate sum dominance. We
extend their results to reduced d-dimensional modular hyperbolas H̄d (a; n) with acoprime to n.
Joint work with Ron Evans (UCSD) and Steven J Miller (Williams).

Jeff Breeding II, Fordham University“The Jacobsthal function and dimensions of spaces of Siegel modular forms”
Abstract: Siegel modular forms can be constructed in many different ways. There
are various lifts either from classical modular forms or from spaces of Jacobi forms
that yield a Siegel modular form with respect to some determined subgroup ofSp(4). Known dimension formulas of the domain spaces of these lifts give a lower
bound for the dimensions of the target space of Siegel modular forms. In this talk,we describe how to obtain an upper bound to dimensions of spaces of paramodular
forms using formal Fourier-Jacobi expansions and the Jacobsthal function. We thencompute precise dimensions for a few cases.
Joint work with Cris Poor and David Yuen.

Javier Cilleruelo, University of Madrid, Spain“Infinite Sidon sequences”
Abstract: We present a method to construct dense infinite Sidon sequences basedon the discrete logarithm.√ We give an explicit construction of an infinite Sidon
sequence B with B(x) = x 2−1+o(1) . Ruzsa proved the existence of a Sidon sequencewith similar counting function but his proof was not constructive. Our method
generalizes√ to B2h -sequences: For all h ≥ 3, there is a Bh -sequence B such that
B(x) = x (h−1) +1−(h−1)+o(1) .

David Covert, University of Missouri - St. Louis“Geometric configurations in Zq ”
Abstract: The Erdős-Falconer distance problem asks one to show that if a set
E ⊂ Fdq is sufficiently large (in the sense of cardinality), then ∆(E) := {(x, y) ∈E × E : (x1 − y1 )2 + · · · + (xd − yd )2 } contains a positive proportion of Fq . Here, Fdq
is the d-dimensional vector space over a finite field with q elements. Likewise, thedot-product problems asks one to show that if E ⊂ Fdq is sufficiently large, then theset Π(E) := {(x, y) ∈ E × E : x1 y1 + · · · + xd yd } contains a positive proportion of
Fq . We discuss both problems in Zdq , where Zq is the ring of integers modulo q. We
obtain nontrivial results in the setting of q = p` , where p is an odd prime. Fourier
analysis and exponential sums play a large role in the proofs.
Joint work with Alex Iosevich and Jonathan Pakianathan

Matthew Devos, Simon Fraser University, Canada“The structure of critical product sets”
Abstract: If A, B are finite nonempty subsets of the (multiplicative) group G, wecall (A, B) critical if |AB| < |A| + |B|. Vosper characterized all critical pairs ingroups of prime order, and Kemperman extended this to arbitrary abelian groups.
Here we describe a further extension to arbitrary groups, and some new corollaries
for critical pairs of the form (A, A).

Mauro Di Nasso, University of Pisa, Italy“Applications in combinatorial number theory of iterated nonstandard extensions
and idempotent ultrafilters”
Abstract: By using nonstandard analysis, and in particular iterated elementary(nonstandard) extensions, we give foundations to a peculiar way of manipulating
idempotent ultrafilters. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of this technique, we give (rather) shortproofs of two important results in combinatorial number theory, namely MillikenTaylor’s Theorem (a generalization of Hindman’s theorem), and Rado’s theorem
about partition regularity of diophantine equations, in a new version formulated interms of idempotent ultrafilters.
Some familiarity with the notion of elementary extension will be assumed in thefirst part of the talk, but in the second part about applications I will not assume
any specific prerequisite (also the notions of ultrafilter and of partition regularity
will be recalled).

Mohamed El Bachraoui, United Arab Emirates University, UAE“A gamma function in two variables”
P∞
1
Abstract: Motivated by the way the Hurwitz zeta function ζ(x, s) = n=0 (n+x)
s
P∞ 1extends the Riemann zeta function ζ(s) = n=1 ns , we introduce a gamma function Γ(x, z) in two complex variables which extends the ordinary gamma function
Γ(z) in the sense that limx→1 Γ(x, z) = Γ(z). We shall give some properties ofΓ(x, z) including functional equations, a multiplication formula, and analogues of
the Stirling formula with asymptotic estimates as consequences.

Leopold Flatto, Bell Labs“The dixie cup problem and the FKG inequality”
Abstract: The coupon collecting problem (also called the dixie cup problem) concerns itself with the number of purchases required on the average to obtain mcomplete sets of n distinct coupons. We first review some of the literature on thisproblem, in particular the paper of Erdős and Renyi, ”A classical problem in probability theory.” One of the main results of that paper is left unproven. In this talkI present a proof which relates the dixie cup problem to the FKG inequality.

George Grossman, Central Michigan University“Combinatorial Identities with Fibonacci and Lucas Numbers”
Abstract: In this paper we show, by exact representation, that each Fibonacci
number can be represented countably many distinct ways, up to sign change, as
triple sum of binomial coefficients. We also find expression for1(L2k+3 Fn + F2k+3 Ln ), k = 0, 1, 2, n ≥ 0,2
and give conjecture if k ≥ 3, n ≥ 0; Ln , Fn are Lucas and Fibonacci numbers,
respectively, such that the expression contains polynomial of degree k in n, anddouble sum of binomial coefficients.
Joint work with Aklilu Zeleke and Xinyun Zhu.

Christopher R. H. Hanusa, Queens College (CUNY)“Applications of simultaneous core partitions: lattice paths, alcoves, and a majorindex”
Abstract: A t-core partition is a partition whose Young diagram has no hooks oflength t. Partitions that are both s-core and t-core for integers s and t are called
simultaneous core partitions. We will discuss the applications of simultaneous core
partitions—we visit with lattice paths, alcoves in a hyperplane arrangement, and a“major index” statistic that recovers a q-analog for Catalan numbers.
Joint work with Brant Jones.

Derrick Hart, Kansas State University“Sumsets of multiplicative subgroups in Zp
Abstract: Let A be a multiplicative subgroup of Zp∗ . Define the k-fold sumset of
A to be kA = {x1 + ... + xk : x1 , ..., xk inA}. Recently, Shkredov has shown that
|2A| >> |A|( 8/5 − ) for |A| < p( 9/17). In this talk we will discuss extending thisresult for |A| < p( 5/9). In addition, we will show that 6A contains Zp∗ f or|A| >p( 33/71 + ).

Kevin Henriot, Université de Montréal, Canada“On arithmetic progressions in A + B + C”
Abstract: We discuss the additive structure of sumsets A + B + C when A, B, C
are dense subsets of a cyclic group. We put particular emphasis in the case where
B and C are quite sparse, since a new technique by Croot and Sisask now allows
one to treat this case. We also discuss the analogous problem in the primes, that is,
the possibility of finding additive structure in A + B + C when A, B, C are subsetsof the primes up to N .

Ginny Hogan, Stanford University“When generalized sumsets are difference dominated”
Abstract: We study the relationship between the number of minus signs in a generalized sumset, A+· · ·+A−· · ·−A, and its cardinality; without loss of generality we
may assume there are at least as many positive signs as negative signs. As addition
is commutative and subtraction is not, we expect that for most A a combination
with more minus signs has more elements than one with fewer; however, recently
Iyer, Lazarev, Miller and Zhang generalized work of Martin and O’Bryant (whodid the case of two summands) and proved that a positive percentage of the time
the combination with fewer minus signs can have more elements. Their analysis
involves choosing sets A uniformly at random from {0, . . . , N }; this is equivalent
to choosing each element of {0, . . . , N } to be in A with probability 1/2. We investigate what happens when instead each element is chosen with probability p(N ),with limN →∞ p(N ) = 0. We prove that the set with more minus signs is larger withprobability 1 as N → ∞ if p(N ) = cN −δ for δ ≥ h−1
h , where h is the number of
total summands in A + · · · + A − · · · − A, and explicitly quantify their relative sizes.
The results generalize earlier work of Hegarty and Miller (who did the h = 2 case),
and we see a phase transition in the behavior of the cardinalities when δ = h−1
h .
Joint work with Steven J Miller.

Jerry Hu, University of Houston - Victoria“The probability that algebraic integers are k-wise relatively prime”
Abstract: We will discuss how to compute the probability that algebraic integers
are k-wise relatively prime. This problem in the case of positive integers has beensolved and the paper will appear in the International Journal of Number Theory(August 2013 issue). The conjectural analogues of k-wise primality for algebraic
varieties over finite fields and schemes over integers will also be discussed.

Renling Jin, College of Charleston“Detailed structure for Freiman’s 3k − 3 theorem”
Abstract: Freiman proved that if |2A| = 2|A| − 1 + b < 3|A| − 3, then 2A contains
an arithmetic progression of length at least 2|A| − 1. We derive a similar result
when |2A| = 3|A| − 3.

Delaram Kahrobaei, New York City Tech (CUNY)“Searching for a secure platform for cryptology: Number theoretic problems vs.
group theoretic problems”
Abstract: In the last decade the idea of using group theoretic problems instead ofnumber theoretic problems has been proposed for secure communication. In thistalk I will speak about some of these new ideas and proposals.

Nathan Kaplan, Harvard University“Higher weight enumerators for codes and algebraic geometry”
Abstract: A linear code C is just a linear subspace of FN
q . The Hamming weight
enumerator of C is a polynomial that keeps track of the number of elements of C
that have i nonzero coordinates for each i ∈ [0, N ]. In this talk we focus on higher
weight enumerators, polynomials that keep track of common zeros of m-tuples of
elements of C. We will explain how to use these higher weight enumerators to study
subcodes of the famous Hamming codes, intersections of two conics in the projective
plane (equivalently, common zeros of homogeneous quadratic polynomials in threevariables), and other codes coming from the evaluation of polynomials. This talkwill be motivated by examples. No prior experience with coding theory will beassumed.

Omar Kihel, Brock University“Recognizing the primes using the permutations”
Abstract: One of the oldest and yet important problem in arithmetic is to recognize
the primes from composites. Wilson proved that an integer n ≥ 2 is a prime number
if and only if n divides (n − 1)! + 1. Lucas provedthat if an integer n is primekthen for every k = 0, . . . , n − 1, we have n−1≡(−1)(mod n). The conversekof Lucas’ theorem has been proved by Cai and Grannville. In the first part ofthis talk, we give a new characterization of the primes using the permutations.

The last part of this talk is devoted to the study of the residue of n−1− (−1)kk
modulo n. This residue is always zero when n is prime. When n is composite it
is of interest to distinguish the values of k for which the residue is zero from thosefor which the residue is nonzero.
It is particularly useful to find explicit
values
of



n−1
k
k
k = k(n) such that n−1
−
(−1)
≡
6
0
(mod
n)
and
gcd
n,
−
(−1)
=
6
1.
k
k
These possible values of k will allow to factorize n. We expected that in the casen odd, k = (nis a good candidate, but for n = 5907 = 3 × 11 × 179, we − 1)/2(n−1)/2
n−1
have (n−1)/2≡ (−1)(mod n). Indeed it is the least value of n, composite,
for which this congruence is satisfied and we found no other value of n composite.
We conjecturethat if n is a product of two odd and distinct prime numbers, thenn−1(n−1)/2≡
6(−1)(mod n). Joint work with Ayad.
(n−1)/2

Sandra Kingan, Brooklyn College (CUNY)“Strong Splitter Theorem”
Abstract: Matroids are a modern type of synthetic geometry where the behaviorof points, lines, planes and higher dimension surfaces are governed by combinatorial axioms. The Splitter Theorem is a central result in matroid theory since it
provides a useful induction tool for structural results. It establishes that (with afew exceptions) if M is a 3-connected matroid with a 3-connected minor N, thenwe can build-up from N to M by a sequence of single-element extensions and coextensions. However, there is no condition on how many extensions may occur before
a coextension must occur. We give a strengthening of the Splitter Theorem, asa result of which, at each step no more than two extensions may occur before acoextension must occur. This sort of stair-stepping approach is the most efficient
way of growing a minor to a matroid.
Joint work with Manoel Lemos.

Thai Hoang Le, University of Texas“Sums of reciprocals of fractional parts”
Abstract: Let kxk denote the distance from x to the nearest integer. We give lower
PN
1, where α, β are real numbers. It turns
and upper bounds for the sum n=1 knαkknβkout that the lower bound is of the correct order of magnitude whenever the pair(α, β) is a counterexample to a notorious conjecture of Littlewood in Diophantine
approximation. We also consider more general sums involving products of many
linear forms in many variables.
Joint work with Jeff Vaaler.

Seva Lev, University of Haifa, Israel“How many solutions can a ± b = 2c have?”
Abstract: For a finite real set C of given size, the number of three-term arithmetic
progressions in C is maximized when C itself is an arithmetic progression. Suppose
now that we split C into two subsets, say A and B, and count only those progressions
with the smallest element in A, and the largest element in B; how many such
”scattered progressions” can there be? Yet more generally, suppose we want tofind two (possibly, intersecting) sets of real numbers, say A and B, with |A| + |B|
prescribed, to maximize the number of three-term progressions with the smallest
element in A, the largest element in B, and the middle element in A ∪ B; howshould A and B be chosen? We answer this and several related questions.
Joint work with Rom Pinchasi.

Neil Lyall, University of Georgia“A purely combinatorial approach to simultaneous polynomial recurrence mod 1”
Abstract: We will present a new (elementary) combinatorial approach to simultaneous quadratic recurrence. Time permitting ,we will also discuss the extension ofthis result to higher degree polynomials and some related new observations concerning polynomial configurations in sumsets. Joint work with Ernie Croot andAlex Rice.

Richard Magner, Eastern Connecticut State University“Ordinary lines and modular hyperbolas”
Abstract: Let S be a finite set of points in Euclidean space. A line that passes
through exactly two distinct points of S is said to be an ordinary line spanned
by S. Let p be an odd prime, m ∈ Z with m ≥ 2, and let Hpm denote themodular hyperbola Hpm = {(x, y) ∈ Z2 : xy ≡ 1(mod pm ), 1 ≤ x, y ≤ pm − 1}.
We prove that the number ofordinary lines spanned by Hpm is bounded below bym−1

6.
pm−1 (p − 1) p 2(p−2) + 13
Joint work with Mizan R. Khan, Steven Senger, and Arne Winterhof.

Steven J. Miller, Williams College“Mind the Gap: Distribution of Gaps in Generalized Zeckendorf Decompositions”
Abstract: Zeckendorf proved that any integer can be decomposed into a unique sumof non-adjacent Fibonacci numbers, Fn . Using continued fractions, Lekkerkerker
showed that the average number of summands in a decomposition of an integer in[Fn , Fn+1 ) is essentially n/(φ2 + 1), where φ is the golden ratio. Miller-Wang generalized this by adopting a combinatorial perspective, proving that for any positive
linear recurrence of the form An = c1 An−1 + c2 An−2 + · · · + cL An+1−L , the numberof summands in decompositions for integers in [An , An+1 ) converges to a Gaussian
distribution as n → ∞.
We prove that the probability of a gap larger than the recurrence length converges to decaying geometrically, with decay rate equal to the largest eigenvalueof the characteristic polynomial of the recurrence, and that the distribution of
the smaller gaps depends on the coefficients of the recurrence (which we analyze
through the combinatorial perspective). These results hold both for the averageover all m ∈ [An , An1 ), as well as holding almost surely for the gap measure associated to individual m. The techniques work for related systems as well, and canalso be used to determine the distribution of the longest gap between summands(which is similar to the distribution of the longest gap between heads in tosses
of a biased coin), as well as for far-difference representations (where positive andnegative summands are allowed).
Joint work with Amanda Bower, Louis Gaudet, Rachel Insoft, Shiyu Li and Phil
Tosteson.

Rishi Nath, York College (CUNY)“Ten facts about self-conjugate partitions”
Abstract: The study of self-conjugate partitions is of interest to number theorists,
representation theorists and combinatorialists. Here we discuss ten facts about
these objects, starting with the work of Frobenius in the early 1900s to recent workby the author and joint work with C. Hanusa.

Mel Nathanson, Lehman College (CUNY)“Cantor packing polynomials in sectors”
Abstract: A packing function on a set Ω in Rn is a one-to-one correspondence
between the set of lattice points in Ω and the set N0 of nonnegative integers. It is
proved that if r and s are relatively prime positive integers such that r divides s−1,then there exist two distinct quadratic packing polynomials on the sector {(x, y) ∈R2 : 0 ≤ y ≤ rx/s}. For the rational numbers 1/s, these are the unique quadratic
packing polynomials. Moreover, quadratic quasi-polynomial packing functions areconstructed for all rational sectors.

Kevin O’Bryant, College of Staten Island (CUNY)“A combinatorial problem arising from Padé approximation”
Abstract: Padé approximation is a critical tool in developing lower bounds on
diophantine approximation of the |21/3 − p/q| > c/q λ sort. This approach beganwith Mahler, and has been developed further by several mathematicians including
D. Chudnovsky, G. Chudnovsky, and M. Bennett. I will outline the general patternof such an attack, and indicate the current weak point in the argument. That weak
point is fundamentally combinatorial in nature, easily understood, and has recently
been solved in joint work with G. Martin and M. Bennett.

Brooke Orosz, Essex County College“A construction of generalized More Sum than Difference Sets”
Abstract: In recent years, there has been interest in finite sets of integers with moresums than differences, that is |2A| > |A − A|. Some articles have also examined
more general questions, such as whether |3A| can be larger than |2A − A|. In fact,
the question has been answered for all cases where the number of sums or differences
is odd. We will address the case where the number of sums or differences is even.
For any positive integer h ≥ 2 and any finite set of integers A, consider S = 2hA
and D = hA − hA. For most sets, |S| ≤ |D|. However, it is possible to construct a
set such that |S| = |D| + 1, which is the union of several arithmetic progressions.

Giorgis Petridis, University of Rochester“Upper bound for higher sumsets with different summands”
Abstract: Let h be a fixed positive integer and A, B1 , . . . , Bh be finite sets in acommutative group. Suppose that |A + Bi | = αi |A|. Then the cardinality of thesumsetA + B1 + · · · + Bh = {a + b1 + · · · + bh : a ∈ A, bi ∈ Bi for all i = 1, . . . , h}is bounded by|A + B1 + · · · + Bh | ≤ α1 . . . αh |A|2−1/h .
The upper bound has the correct dependence in |A| and the αi , yet not in h.
It is also of interest to note that there are several ways to prove this inequality,
using graph theory, projections and entropy. We explain how the graph-theoretic
approach (developed by Imre Ruzsa) can be modified to yield
α1 . . . αh|A|2−1/h ,|A + B1 + · · · + Bh | ≤ Chfor an absolute constant C. This improved upper bound is, up to a constant, sharp
in all the parameters. This is one of the few instances where the correct dependence
in all parameters is known for similar sumset-related problems.
Joint work with Brendan Murphy and Eyvi Palsson.

Alex Rice, Bucknell University“Squares and primes in generalized arithmetic progressions”
Abstract: A classical question in analytic number theory is, given an arithmetic
progression which has any business containing primes (i.e. the starting point iscoprime to the step size), what is the smallest prime in that progression? The best
known results fall under the banner of Linnik’s Theorem. One could ask a similar
question replacing primes with perfect squares, in which case it becomes completely
trivial. However, if one expands from progressions to generalized arithmetic progressions, the question becomes much less trivial in the squares case and much lessclassical in the primes case. Here we discuss some results of this type.
Joint work with Ernie Croot and Neil Lyall.

Tom Sanders, Oxford University, UK“The structure theory of set addition revisited”
Abstract: In this series of talks we shall look to introduce some of the more recent
developments in the structure theory of set addition. In the first we shall cover someof the basic tools including Petridis’ wonderful work on Plünnecke’s inequality. The
second two talks will cover Freı̆man’s theorem, splitting it into a combinatorial andharmonic analytic part.

Steven Senger, University of Delaware“A multi-scale approach to the Erdős-Falconer type single distance problems”
Abstract: We discuss the Erdős-Falconer type single distance problems, and present
new estimates, in some special cases, on the measure of the set of pairs of points
from a subset of [0, 1]d which are -nearly a given distance apart.

Satyanand Singh, New York City Tech (CUNY)“Exponential diophantine equations of Conway and Nathanson”
Abstract: A demonstration will be made of the insolvability of the diophantine
equations |2a − 3b | = 149 and |2c − 3d | = 151. We will show this in two ways:
The first by finding obstructions in the rings Z/mZ, where m ∈ N0 , and the otherby using a result that was derived from Baker’s method. This in turn will help
establish that λ2,3 (4) = 150, where λ2,3 (h) plays an important role in the studyof geometric diameter and additive number theory. A discussion will be made ofrelated open problems and additional results.

Jonathan Sondow, New York
“Ramanujan, Robin, the Riemann Hypothesis, and recent results”
Abstract: First I will tell the story of how the hidden part of Ramanujan’s 1915
thesis “Highly composite numbers” was discovered in 1987 by Jean- Louis Nicolas at the Ramanujan 100 Conference held in Urbana, IL. Then I will explain thepart on an asymptotic upper bound for the sum-of- divisors function, which Ramanujan proved assuming the Riemann Hypothesis (RH). Next I will state the
non-asymptotic version equivalent to RH proved in 1984 by Guy Robin, whose thesis advisor was Nicolas. Finally I will describe recent related criteria for RH proved
jointly with Geoffrey Caveney and Nicolas. See http://arxiv.org/abs/1211.6944 forlecture slides (the last one is surprising). My paper will appear in the Proceedingsof the Ramanujan 125 Conference held in Gainesville, FL in 2012.

Dmitry Zhelezov, Chalmers Institute of Technology, Sweden“Product sets cannot contain long arithmetic progressions”
Abstract: Let B be a set of real numbers of size n. We prove that the length of thelongest arithmetic progression containedin the product set B.B = {bi bj | bi , bj ∈ B}√
cannot be greater than O(n1+1/ log log n ) and present an example of a product set
containing an arithmetic progression of length Ω(n log n), so the obtained upperbound is close to the optimal.