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 K 2 r K 4 | 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.