Sets with few differences in abelian groups

TL;DR

This paper investigates the minimal size of difference sets in abelian groups, proposing a conjecture, proving it for specific cases, and resolving a related conjecture on signed sumsets.

Contribution

It introduces a conjecture for the minimal difference set size in abelian groups and proves it for cyclic groups and finite field vector spaces, advancing understanding of sumset differences.

Findings

Conjecture for minimal difference set size proposed.

Proved conjecture for cyclic groups.

Confirmed conjecture for vector spaces over finite fields.

Abstract

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sumsets.