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

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.