Finer control on relative sizes of iterated sumsets

TL;DR

This paper demonstrates that in any infinite abelian group, one can construct finite sets with prescribed differences in sizes of their iterated sumsets, and explores minimal size and diameter questions for such sets.

Contribution

It establishes the existence of finite sets with specified sumset size differences in infinite abelian groups and initiates study on their minimal cardinalities and diameters.

Findings

Existence of finite sets with prescribed sumset size differences.

Initial results on minimal sizes and diameters of such sets.

Abstract

Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also raise, and begin to address, questions about the smallest possible cardinalities and diameters of such sets $A,B$.