Relative sizes of iterated sumsets

TL;DR

This paper constructs finite subsets of integers with prescribed relative sizes of their iterated sumsets, resolving a problem of Nathanson and extending to other abelian groups and equality conditions.

Contribution

It provides a method to realize any prescribed order of sumset sizes across multiple subsets and iterated sums, solving Nathanson's problem and generalizing to broader settings.

Findings

Existence of subsets with prescribed sumset size orderings

Extension to arbitrary infinite abelian groups

Ability to specify equalities among sumset sizes

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 $\sigma_1, \ldots, \sigma_H \in \mathfrak{S}_n$, there exist finite subsets $A_1, \ldots, A_n \subseteq \mathbb{Z}$ such that for each $1 \leq h \leq H$, the relative order of the quantities $|h A_1|, \ldots, |h A_n|$ is given by $\sigma_h$. 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.