On Lev's periodicity conjecture

We classify the sum-free subsets of ${\mathbb{F}}_3^n$ whose density exceeds $\frac{1}{6}$. This yields a resolution of Vsevolod Lev’s periodicity conjecture, which asserts that if a sum-free subset $A\subseteq {\mathbb{F}}_3^n$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A+v=A$), then $\mathrm{\mid }A\mathrm{\mid }\le \frac{1}{2}(3^{n-1}+1)$ – a bound known to be optimal if $n\mathrm{\ne }2$, while for $n=2$ there are no such sets.