Abstract
As an easy corollary of a well-known theorem of Kneser, if A is a subset of the elementary abelian group Z5n of density 5−n|A|>0.4, then 3A=Z5n. We establish the complementary stability result: if 5−n|A|>0.3 and 3A≠Z5n, then A is contained in a union of two cosets of an index-5 subgroup of Z5n. Here the density bound 0.3 is sharp.Our argument combines a combinatorial reasoning with a somewhat non-standard application of the character sum technique.
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