UC San Diego

Let G be a finite abelian group of torsion r and let A be a subset of G. The Freiman-Ruzsa theorem asserts that if |A + A| ≤ K|A| then A is contained in a coset of a subgroup of G of size at most K2rK4|A|. It was conjectured by Ruzsa that the subgroup size can be reduced to ^{r C K}|A| for some absolute constant C ≥ 2. This conjecture was verified for r = 2 in a sequence of recent works, which have, in fact, yielded a tight bound. In this work, we establish the same conjecture for any prime torsion. © 2014 Elsevier Inc.