Iterated Sumsets and Subsequence Sums

Let $G\cong \mathbb{Z}\mathrm{/}{m}_1\mathbb{Z}\times \dots \times \mathbb{Z}\mathrm{/}{m}_r\mathbb{Z}$ be a finite abelian group with $m_1\mid \dots \mid m_r=\exp (G)$. The Kemperman Structure Theorem characterizes all subsets $A,B\subseteq G$ satisfying $\mathrm{\mid }A+B\mathrm{\mid }<\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }$ and has been extended to cover the case when $|A+B|\leq |A|+|B|$. Utilizing these results, we provide a precise structural description of all finite subsets $A\subseteq G$ with $\mathrm{\mid }nA\mathrm{\mid }\le (\mathrm{\mid }A\mathrm{\mid }+1)n-3$ when $n\ge 3$ (also when $G$ is infinite), in which case many of the pathological possibilities from the case $n=2$ vanish, particularly for large $n\geq \exp(G)-1$. The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence $S$ of terms from $G$ having length $\mathrm{\mid }S\mathrm{\mid }\ge 2\mathrm{\mid }G\mathrm{\mid }-1$ must either have every element of $G$ representable as a sum of $\mathrm{\mid }G\mathrm{\mid }$-terms from $S$ or else have all but $\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }-2$ of its terms lying in a common $H$-coset for some $H\le G$. We show that the much weaker hypothesis $\mathrm{\mid }S\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }+\exp (G)$ suffices to obtain a nearly identical conclusion, where for the case $H$ is trivial we must allow all but $\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }-1$ terms of $S$ to be from the same $H$-coset. The bound on $\mathrm{\mid }S\mathrm{\mid }$ is improved for several classes of groups $G$, yielding optimal lower bounds for $\mathrm{\mid }S\mathrm{\mid }$. We also generalize Olson’s result for $\mathrm{\mid }G\mathrm{\mid }$-term subsums to an analogous one for $n$-term subsums when $n\ge \exp (G)$, with the bound likewise improved for several special classes of groups. This improves previous generalizations of Olson’s result, with the bounds for $n$ optimal.