Asymptotic Combinatorics, Stability and Ultrafilters - EVENTO ANNULLATO

Data Seminario: 
Ora Inizio: 
Ora Fine: 
Amador Martin-Pizzarro
Albert-Ludwigs-Universität Freiburg

A finite subset A of a group G is said to have doubling K if
the set A\cdot A consisting of products a\cdot b, with a and b in A has
size at most K|A|. Extreme examples of sets with small doubling are
cosets of subgroups.

Theorems of Freiman-Ruzsa type assert that sets with small doubling are
"not too far" from being subgroups. Freiman's original theorem asserts
that a finite subset of the integers with small doubling is efficiently
contained in a generalized arithmetic progression.  A version of this
result for abelian groups of bounded exponent was given by Ruzsa:  a
finite subset with small doubling K of an abelian group G of exponent r
is contained in a subgroupH of G of size bounded by K, r and |A| (but
the bound he exhibited is exponential). A natural reformulation of the
problem is the polynomial Freiman-Ruzsa conjecture, one of the central
open problems in additive combinatorics, which aims to find polynomial
bounds (in K) so that any subset A of small doubling K in an
infinite-dimensional vector space over F_2 can be covered by finitely
many translates of some subspace, whose size is commensurable to the
size of A. Improvements of this result have been subsequently obtained
by many authors for arbitrary (possibly infinite and non-abelian)

Motivated by work of  E. Hrushovski, we will present an on-going work
with D. Palacin (Freiburg) and J. Wolf (Cambridge) of Freiman-Ruzsa type
under some assumptions of model-theoretic nature.