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 $|A+A| \le K|A|$ then $A$ is contained in a coset of a subgroup of size at most $K^2 r^{K^4} |A|$. Ruzsa conjectured that the bound can be improved to $r^{cK} |A|$ for some absolute constant $c \ge 2$. This conjecture was verified for $r=2$ in a sequence of recent works. In this work, we establish the same conjecture for any prime torsion. Joint work with Chaim Even-Zohar.
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