Relative sizes of iterated sumsets

Abstract

Let hA denote the h-fold sumset of a subset A of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations ${\sigma }_1,\dots ,{\sigma }_H\in S_n$, there exist finite subsets $A_1,\dots ,A_n\subseteq Z$ such that for each $1\le h\le H$, the relative order of the quantities $\left|h{A}_1\right|,\dots ,\left|h{A}_n\right|$ is given by ${\sigma }_h$. We also establish extensions where $Z$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.