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 (finite) 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 on-going work with D. Palacin (Freiburg) and J. Wolf (Cambridge) of Freiman-Ruzsa type under stability, an assumption of model-theoretic nature.