A note on iterated sumsets races

TL;DR

This paper explores the behavior of differences in sizes of iterated sumsets, showing that for any number of sign changes, there exist finite sets with identical diameters exhibiting this property.

Contribution

It demonstrates that for any integer n, finite sets with equal diameters can have their sumset size differences change signs at least n times, addressing a question by Nathanson.

Findings

Constructed sets with prescribed sign change behavior

Connected sumset size differences to combinatorial properties

Serves as a foundation for further research in sumset dynamics

Abstract

This short note answers a question raised by Nathanson \cite{Nath25} about "races" between iterated sumsets. We prove that for any integer $n$, there are finite sets of integers $A$ and $B$ with same diameter such that the signs of the elements of the sequence $(|hA|-|hB|)_h$ changes at least $n$ times. Kravitz proved in \cite{Kravitz} a much better result. This brief and modest note may serve as a stepping stone towards his work.