Iterated Sumsets and Olson's Generalization of the Erdős-Ginzburg-Ziv Theorem

Let $G\cong Z/m_{1}Z\times \dots \times Z/m_{r}Z$ be a finite abelian group with $m_1|\dots |m_r=\exp \left(G\right)$. The Kemperman Structure Theorem characterizes all subsets $A,B\subseteq G$ satisfying $|A+B|<\left|A\right|+\left|B\right|$ and has been extended to cover the case when $|A+B|\le \left|A\right|+\left|B\right|$. Utilizing these results, we provide a precise structural description of all finite subsets $A\subseteq G$ with $\left|nA\right|\le \left(\right|A|+1)n-3$ 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 $n\ge \exp \left(G\right)-1$. 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 $\left|S\right|\ge 2\left|G\right|-1$ must either have every element of G representable as a sum of $\left|G\right|$-terms from S or else have all but $|G/H|-2$ of its terms lying in a common H-coset for some $H\le G$. We show that the much weaker hypothesis $\left|S\right|\ge \left|G\right|+\exp \left(G\right)$ suffices to obtain a nearly identical conclusion, where for the case H is trivial we must allow all but $|G/H|-1$ terms of S to be from the same H-coset. The bound on $\left|S\right|$ is improved for several classes of groups G, yielding optimal lower bounds for $\left|S\right|$.