## The workshop will take place at The Gilman Building, room 326, Tel Aviv University:

# Schedule

** July 12 **

09:30-10:00: Coffee
10-10:45: Oriol Serra
11-11:45: Melvyn Nathanson
12:00-13:30: Lunch
14:00-14:45: Yonutz Stanchescu
15:00-15:45: Mercede Maj and Patrizia Longobardi
15:45-16:15: Coffee
16:15-17:00: Personal comments

** July 13 **

09:30-10:00: Coffee
10-10:45: Endre Szemerèdi
11-11:45: Vsevolod Lev
12:00-13:30: Lunch
14:00-14:45: Jean-Marc Deshouillers
15:00-15:45: Emmanuel Breuillard
15:45-16:15: Coffee
16:15-16:45: Gregory Freiman and closing remarks

# List of speakers

Emmanuel Breuillard, Munster
Jean-Marc Deshouillers, Bordeaux
Vsevolod Lev, Haifa
Patrizia Longobardi, Salerno
Mercede Maj, Salerno
Melvyn Nathanson, New York
Oriol Serra, Barcelona
Yonutz Stanchescu, Tel Aviv
Endre Szemerèdi, Budapest
# Titles and Abstracts

** Oriol Serra, ** Universitat Politecnica de Catalunya,
Barcelona, Spain

**
Title: Doubling and Volume: A conjecture of Freiman
**
** Abstract: **
By the well-known Freiman-Ruzsa theorem, a set A of integers with
small doubling, |A+A| \leq c|A|, is a dense subset of a multidimensional
arithmetic progression. The search for efficient quantitative estimations
of the density of A in the progression in terms of the doubling constant
c has been the object of intense research by several authors, including
Freiman, Ruzsa, Bilu, Chang, Sanders and Konyagin. Schoen obtained an
asymptotic bound which is essentially best possible.

An approach to the problem is to determine the maximum volume of a
set with given cardinality and doubling. The dimension of A is the
largest dimension of an integer lattice containing a Freiman isomorphic
copy of A not contained in a hyperplane. The additive volume of a
d-dimensional set A is defined as the smallest cardinality of the
convex hull of sets isomorphic to A in the d-dimensional lattice.
Freiman conjectured in 2008 a formula for the value of this maximum volume.

The talk will discuss recent advancements in this direction, in
particular proving the conjecture for the class of so-called
chains. Some extensions and implications will be also discussed.

Joint work with G. A. Freiman.

** Melvyn Nathanson, ** CUNY,
New York, USA

**
Title: Every finite subset of an abelian group is an asymptotic
approximate group
**
** Abstract: **
If $A$ is a nonempty subset of an additive abelian group $G$, then
the $h$-fold sumset is

hA = \{x_1 + \cdots + x_h : x_i \in A_i \text{ for } i=1,2,\ldots, h\}.

We do not assume that $A$ contains the identity, nor that
$A$ is symmetric, nor that $A$ is finite. The set $A$ is an
$(r,\ell)$-approximate group in $G$ if there exists a subset $X$
of $G$ such that $|X| \leq \ell$ and $rA \subseteq XA$. The set
$A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$
is an $(r,\ell)$-approximate group for all sufficiently large $h$.

It is proved that every polytope in a real vector space is an
asymptotic $(r,\ell)$-approximate group, that every finite set of
lattice points is an asymptotic $(r,\ell)$-approximate group, and
that every finite subset of every abelian group is an asymptotic
$(r,\ell)$-approximate group.

Preprints are posted on arXiv: 1512.03130 and 1512.06478.

** Yonutz Stanchescu, ** Afeka-Tel Aviv Academic
College, Israel

**
Title: Small doubling in Baumslag-Solitar groups and dilates of
integers
**
** Abstract: **
We will present a connection between abelian sumsets estimates and
growth in non-abelian groups. More precisely, we investigate small
doubling problems in Baumslag-Solitar groups using new direct and
inverse results for sums of dilates of integers.

Joint work with G. A. Freiman, M. Herzog, P. Longobardi and M.
Maj.

** Mercede Maj and Patrizia Longobardi, ** University
of Salerno, Italy

**
Title: On some Freiman's problems concerning products of subsets
in groups
**
** Abstract: **
Let G denote an arbitrary group. If S is a subset of G,
we define its square S^2 by

S^2={ x_1x_2 : x_1,x_2 in S}.

In this talk we are interested in two problems, proposed by Gregory
Freiman, concerning products of finite subsets in groups.

First we investigate the class DS(m), with m > 1 an integer,
where a group G belongs to DS(m)
if the cardinality of S^2 is less that m^2 for every subset S
of G of order m.

Then we will present some recent results, obtained jointly with
G.A. Freiman, M. Herzog, Y.V. Stanchescu, A. Plagne and D.J.S.
Robinson concerning the structure of finite subsets S of an
orderable group satisfying the inequality:

|S^2| \leq \alpha|S| + \beta, with \alpha = 3 and small
values of |\beta|.

** Endre Szemerèdi, ** Renyi Institute,
Budapest, Hungary

**
Title: Applications of the fundamental theorem of Freiman
**
** Abstract: **
We are going to discuss some applications of Freiman's theorem in
subset-sum problems. We are going to give estimations and
structural results.

** Vsevolod Lev, ** University of Haifa, Israel

**
Title: Quadratic residues and difference sets
**
** Abstract: **
It has been conjectured by Sarkozy that with finitely many
exceptions, the set of quadratic residues modulo a prime $p$ cannot
be represented as a sumset $\{a+b\colon a\in A, b\in B\}$ with
non-singleton $A,B\subset F_p$. The case $A=B$ of this conjecture has
been established by Shkredov. The analogous problem for differences
remains open: is it true that for all sufficiently large primes
$p$, the set of quadratic residues modulo $p$ is not of the form
$\{a'-a''\colon a',a''\in A,\,a'\ne a''\}$ with $A\subset F_p$?

We present the results of our recent paper, joint with Jack
Sonn, where a presumably more tractable variant of this problem is
considered: is there a set $A\subset F_p$ such that every quadratic
residue has a *unique* representation as $a'-a''$ with $a',a''\in A$,
and no non-residue is representable in this form? We give a number of
necessary conditions for the existence of such $A$, involving for the
most part the behavior of primes dividing $p-1$. These conditions
enabled us to rule out all primes $p$ bigger than 13 and smaller
than $10^{20}$
(the primes $p=5$ and $p=13$ being conjecturally the only exceptions).

** Jean-Marc Deshouillers, **
University of Bordeaux, France

**
Title: Probabilistic methods for Waring's problem: old and
new results
**
** Abstract: **
TBA

** Emmanuel Breuillard, **
Universitat Munster, Germany

**
Title: The Elekes-Szabo theorem in higher dimension
**
** Abstract: **
I will discuss the Elekes-Szabo theorem regarding expanding
polynomials and its higher dimensional generalization in which a group
structure is derived from a combinatorial constraint. In joint work
with Hong Wang we determine the possible groups arising this way.

# Organizing Committee

Noga Alon, Jean-Marc Deshouillers, Marcel Herzog, Vsevolod F. Lev,
Melvyn Nathanson, Alain Plagne, Oriol Serra, Yonutz Stanchescu

The Baumritter Chair of Combinatorics and Computer Science

The Communication Coding Theory Chair