The Freiman-Ruzsa theorem over finite fields

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 rCK|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.