Large sum-free sets in finite vector spaces I

Let p be a prime number with p≡ 23 and let n≥ 1 be a dimension. It is known that a sum-free subset of Fₚⁿ can have at most the size 13(p+1)pⁿ⁻¹ and that, up to automorphisms of Fₚⁿ, the only extremal example is the `cuboid' [p+13, 2p-13]× Fₚⁿ⁻¹. For p≥ 11 we show that if a sum-free subset of Fₚⁿ is not contained in such an extremal one, then its size is at most 13(p-2)pⁿ⁻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≡ 13 it is less interesting.