The structure of large sum-free sets in ${\mathbb{F}}_p^n$

A set $A\subset {\mathbb{F}}_p^n$ is sum-free if $A+A$ does not intersect $A$. If $p\equiv 2\mathrm{m}\mathrm{o}\mathrm{d}3$, the maximal size of a sum-free in ${\mathbb{F}}_p^n$ is known to be $(p^n+p^{n-1})\mathrm{/}3$. We show that if a sum-free set $A\subset {\mathbb{F}}_p^n$ has size at least $p^n\mathrm{/}3-p^{n-1}\mathrm{/}6+p^{n-2}$, then there exists subspace $V<{\mathbb{F}}_p^n$ of co-dimension 1 such that $A$ is contained in $(p+1)\mathrm{/}3$ 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.