On Lev's periodicity conjecture

TL;DR

This paper classifies large sum-free subsets in finite vector spaces over F_3, resolving Lev's periodicity conjecture by establishing bounds on maximal aperiodic sum-free sets.

Contribution

It provides a complete classification of sum-free subsets exceeding a certain density and proves Lev's conjecture on the maximal size of aperiodic sum-free sets.

Findings

Classified sum-free subsets with density > 1/6 in F_3^n

Resolved Lev's periodicity conjecture for all n

Established optimal bounds for maximal aperiodic sum-free sets

Abstract

We classify the sum-free subsets of ${\mathbb F}_3^n$ whose density exceeds $\frac16$. 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 $|A|\le \frac12(3^{n-1}+1)$ -- a bound known to be optimal if $n\ne 2$, while for $n=2$ there are no such sets.