A note on iterated sumsets races

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 $(\mathrm{\mid }hA\mathrm{\mid }-\mathrm{\mid }hB\mathrm{\mid }{)}_h$ 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.