On Lev's periodicity conjecture

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.

Theorems & Definitions (50)

• Definition 1.1
• Definition 1.2
• Lemma 1.3
• Theorem 1.4
• Corollary 1.5: Lev's periodicity conjecture
• Lemma 2.1
• proof
• Lemma 2.2: Lev
• proof
• proof : Proof of Lemma \ref{['lem:13']}