Stability result for sets with $3A\ne{\mathbb Z}_5^n$

TL;DR

This paper proves a sharp density threshold for subsets of ${f Z}_5^n$ where the sumset $3A$ does not cover the entire group, showing such sets are contained within two cosets of a subgroup.

Contribution

It establishes a sharp stability result for subsets of ${f Z}_5^n$ with density above 0.3, characterizing their structure when $3A$ is not the whole group.

Findings

Density threshold of 0.3 is sharp for the stability result.

Sets with density >0.3 and $3A eq {f Z}_5^n$ are contained in two cosets.

The proof combines combinatorial and character sum techniques.

Abstract

As an easy corollary of Kneser's Theorem, if $A$ is a subset of the elementary abelian group ${\mathbb Z}_5^n$ of density $5^{-n}|A|>0.4$, then $3A={\mathbb Z}_5^n$. We establish the complementary stability result: if $5^{-n}|A|>0.3$ and $3A\ne{\mathbb Z}_5^n$, then $A$ is contained in a union of two cosets of an index-$5$ subgroup of ${\mathbb Z}_5^n$. Here the density bound $0.3$ is sharp. Our argument combines combinatorial reasoning with a somewhat non-standard application of the character sum technique.