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)

groups.

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.