On Lev's periodicity conjecture

We classify the sum-free subsets of F₃ⁿ whose density exceeds 16. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset A⊆ F₃ⁿ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector v satisfying $A+v=A$), then |A|≤ 12(3ⁿ⁻¹+1) -- a bound known to be optimal if n≠ 2, while for $n=2$ there are no such sets.