Sum-free sets in $Z_5^n$

Abstract

It is well-known that for a prime $p\equiv 2(\mathrm{mod}3)$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group ${\mathbb{Z}}_p^n$ is $\frac{1}{3}(p+1)p^{n-1}$. We establish a matching stability result in the case $p=5$: if $A\subseteq {\mathbb{Z}}_5^n$ is a sum-free subset of size $|A|>\frac{3}{2}\cdot 5^{n-1}$, then there are a subgroup $H<{\mathbb{Z}}_5^n$ of size $|H|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq (e+H)\cup (-e+H)$.