The Structure of Large SUM-Free Sets in 𝔽n p

Abstract

ABSTRACT A set $A\subset \mathbb{F}_p^n$ is sum-free if A + A does not intersect A. If $p\equiv 2 \;\mathrm{mod}\; 3$, the maximal size of a sum-free set in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset \mathbb{F}_p^n$ has size at least $p^n/3-p^{n-1}/6+p^{n-2}$, then there exists subspace $V\lt\mathbb{F}_p^n$ of codimension 1 such that A is contained in $(p+1)/3$ cosets of V. For p = 5 specifically, we show the stronger result that every sum-free set of size larger than $1.2\cdot 5^{n-1}$ has this property, thus improving on a recent theorem of Lev.