Georgia Institute of Technology College of Sciences

The Balog-Szemeredi-Gowers theorem is a widely used tool in additive combinatorics, and it says, roughly, that if one has a set A such that the sumset A+A is "concentrated on few values," in the sense that these values v each get close to n representations as v = a+b, with a,b in A, then there is a large subset A' of A such that the sumset A'+A' is "small" -- i.e. it has size a small multiple of n. Later, Sudakov, Szemeredi and Vu generalized this result to handle multiple sums A_1 + ... + A_k. In the present talk we will present a refinement of this result of Sudakov, Szemeredi and Vu, where we get better control on the growth of sums A'+...+A'. This is joint work with Ernie Croot.