Stability result for sets with 3A≠\mathbb Z₅ⁿ

2016/02/22 by Lev, Vsevolod F.
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1602.06715

Typo fixed, otherwise identical to the previous version

Abstract

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

Related

• Sum-free sets in Z₅ⁿ
• Infinite unrestricted sumsets in subsets of abelian groups with large density
• On a Theorem of Deshouillers and Freiman
• Iterated Sumsets and Setpartitions
• On the number of sets with a given doubling constant