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 $\ddot{I}{ƒ}_1,\dots ,\ddot{I}{ƒ}_H\in {\mathfrak{S}}_n$, there exist finite subsets $A_1,\dots ,A_n\subseteq \mathbb{Z}$ such that for each $1\le h\le H$, the relative order of the quantities $|h{A}_1|,\dots ,|h{A}_n|$ is given by $\ddot{I}{ƒ}_h$. We also establish extensions where $\mathbb{Z}$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.

Substantially strengthened the main result