  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.
