Large sum-free sets in finite vector spaces II

TL;DR

This paper characterizes large sum-free sets in finite vector spaces over , showing they are either contained in two hyperplanes or have a specific product structure.

Contribution

It proves a classification of large sum-free sets in ^n, answering a question posed by Versteegen and identifying their precise structure.

Findings

Large sum-free sets are contained in two hyperplanes or areomorphic to a specific product set.

Established a size threshold of 28 ^{n-3} for the classification.

Connected the structure of sum-free sets to a known example by Lev and Versteegen.

Abstract

Answering a question of Leo Versteegen, we prove that for $n\ge 3$ every sum-free set $A\subseteq\mathbb{F}_5^n$ with $|A|\ge 28\cdot 5^{n-3}$ is either contained in the union of two parallel hyperplanes, or isomorphic to $\Lambda\times \mathbb{F}_5^{n-3}$, where $\Lambda\subseteq \mathbb{F}_5^3$ denotes a certain sum-free set of size $28$ discovered by Vsevolod Lev and Leo Versteegen.