Relative sizes of iterated sumsets

Noah Kravitz

Published 2024-12-24Version 1

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Nathanson asked if there exist finite sets $A,B \subseteq \mathbb{Z}$ and natural numbers $h_1<h_2<h_3$ such that $|h_1A|<|h_1B|$, $|h_2B|<|h_2A|$, and $|h_3A|<|h_3B|$. We answer this question in the affirmative and establish a generalization with arbitrarily many sets and arbitrarily many values of $h$.