Large sum-free sets in finite vector spaces I

Let $p$ be a prime number with $p\equiv 2\pmod{3}$ and let $n\ge 1$ be a dimension. It is known that a sum-free subset of ${\mathbb F}_p^n$ can have at most the size $\frac13(p+1)p^{n-1}$ and that, up to automorphisms of ${\mathbb F}_p^n$, the only extremal example is the `cuboid' $\bigl[\frac{p+1}3, \frac{2p-1}3\bigr]\times {\mathbb F}_p^{n-1}$. For $p\ge 11$ we show that if a sum-free subset of ${\mathbb F}_p^n$ is not contained in such an extremal one, then its size is at most $\frac13(p-2)p^{n-1}$. This bound is optimal and we classify the extremal configurations. The remaining cases $p=2, 5$ are known to behave differently. For $p=3$ the analogous question was solved by Vsevolod Lev, and for $p\equiv 1\pmod{3}$ it is less interesting.