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 $p\equiv 2 \mod 3$, the maximal size of a sum-free 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 $V<\mathbb{F}_p^n$ of co-dimension 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.