The Haight-Ruzsa method for sets with more differences than multiple sums

TL;DR

This paper discusses the Haight-Ruzsa method for constructing subsets of additive groups where the difference set is large but the multiple sumset is small, extending the method to other groups.

Contribution

It describes and modestly extends the Haight-Ruzsa construction to create sets with more differences than multiple sums in various additive abelian groups.

Findings

Constructed sets with large difference sets and small multiple sumsets.

Extended the Haight-Ruzsa method to other additive groups.

Abstract

Let $h$ be a positive integer and let $\varepsilon > 0$. The Haight-Ruzsa method produces a positive integer $m^*$ and a subset $A$ of the additive abelian group $\mathbf{Z}/m^*\mathbf{Z}$ such that the difference set is large in the sense that $A-A = \mathbf{Z}/m^*\mathbf{Z}$ and $h$-fold sumset is small in the sense that $|hA| < \varepsilon m^*$. This note describes, and in a modest way extends, the Haight-Ruzsa argument, and constructs sets with more differences than multiple sums in other additive abelian groups.