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As an easy corollary of Kneser's Theorem, if $A$ is a subset of theelementary 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 oftwo cosets of an index-$5$ subgroup of ${\mathbb Z}_5^n$. Here the densitybound $0.3$ is sharp.  Our argument combines combinatorial reasoning with a somewhat non-standardapplication of the character sum technique.

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