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ABSTRACT 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+p^n-1) /3. We show that if a sum-free set A Fₚⁿ has size at least pⁿ/3-p^n-1/6+p^n-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 5^n-1 has this property, thus improving on a recent theorem of Lev.
Leo Versteegen (Fri,) studied this question.