The Structure of Large SUM-Free Sets in 𝔽n p

A set A⊂ Fₚⁿ is sum-free if A + A does not intersect A. If p≡ 2 \;mod\; 3, the maximal size of a sum-free set in Fₚⁿ is known to be (pⁿ+pⁿ⁻¹)/3. We show that if a sum-free set A⊂ Fₚⁿ has size at least pⁿ/3-pⁿ⁻¹/6+pⁿ⁻², then there exists subspace Vₚⁿ 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· 5ⁿ⁻¹ has this property, thus improving on a recent theorem of Lev.