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+AKA then A is contained in a coset of a subgroup of G of size at most K2rK4A . It was conjectured by Ruzsa that the subgroup size can be reduced to rCK for some absolute constant C2. 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.
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