Sum-free sets in $Z_5^n$

TL;DR

This paper proves a stability result for large sum-free subsets in the group b Z_5^n, showing they are contained within specific structured subsets when exceeding a certain size threshold.

Contribution

It establishes a stability theorem for sum-free sets in b Z_5^n, characterizing their structure when they are sufficiently large.

Findings

Sum-free sets larger than d 1.5 d 5^{n-1} are structurally contained within specific cosets.

Characterization of large sum-free sets in b Z_5^n.

Matching bounds for maximum size and structure of sum-free sets.

Abstract

It is well-known that for a prime $p\equiv 2\pmod 3$ and integer $n\ge 1$, the maximum possible size of a sum-free subset of the elementary abelian group $\mathbb Z_p^n$ is $\frac13\,(p+1)p^{n-1}$. 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 $|A|>\frac32\cdot5^{n-1}$, then there are a subgroup $H<\mathbb Z_5^n$ of size $|H|=5^{n-1}$ and an element $e\notin H$ such that $A\subseteq(e+H)\cup(-e+H)$.