UBC Discrete Math Seminar: Ilya Shkredov

We develop the theory of the additive dimension dim(A), i.e. the size of a maximal dissociated subset of a set A. It was shown that the additive dimension is closely connected with the growth of higher sumsets nA of our set A. We apply this approach to demonstrate that for any small multiplicative subgroup Γ the sequence |nΓ| grows very fast. Also, we obtain a series of applications to the sum--product phenomenon and to the Balog--Wooley decomposition--type results.