Small asymmetric sumsets in elementary abelian 2-groups

TL;DR

This paper investigates the structure of sumsets in elementary abelian 2-groups, showing that small sumsets are either the whole group or nearly so, with detailed conditions and refinements based on subset sizes.

Contribution

It extends previous results by characterizing when sumsets are large or nearly the entire group in elementary abelian 2-groups, including new bounds and conditions.

Findings

If |A+B|<|A|+|B|, then A+B=G or its complement is contained in a coset of a subgroup of index at least 8.

The size of A+B is at least 7/8 of |G| when the sumset is not the whole group.

Conditions are provided for when the sumset containment is strict and for cases with significantly different subset sizes.

Abstract

Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then either A+B=G, or the complement of A+B in G is contained in a coset of a subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate conditions for the containment to be strict, and establish a refinement in the case where the sizes of A and B differ significantly.