Stability result for sets with $3A\ne{\mathbb Z}_5^n$

Abstract: As an easy corollary of Kneser's Theorem, if $A$ is a subset of the elementary abelian group ${\mathbb Z}_5^n$ of density $5^{-n}|A|>0.4$, then $3A={\mathbb Z}_5^n$. We establish the complementary stability result: if $5^{-n}|A|>0.3$ and $3A\ne{\mathbb Z}_5^n$, then $A$ is contained in a union of two cosets of an index-$5$ subgroup of ${\mathbb Z}_5^n$. Here the density bound $0.3$ is sharp. Our argument combines combinatorial reasoning with a somewhat non-standard application of the character sum technique.