Iterated Sumsets and Subsequence Sums

TL;DR

This paper characterizes the structure of subsets and sequences in finite abelian groups with specific sumset and subsequence sum properties, extending classical theorems and improving bounds for various group classes.

Contribution

It provides a precise structural description of subsets with small multiple sumsets and generalizes Olson's subsequence sum results with optimal bounds.

Findings

Structural description of subsets with small n-fold sumsets

Generalization of Olson's subsequence sum theorem with weaker hypotheses

Improved bounds for specific classes of groups

Abstract

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The Kemperman Structure Theorem characterizes all subsets $A,\,B\subseteq G$ satisfying $|A+B|<|A|+|B|$ 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 $|nA|\leq (|A|+1)n-3$ when $n\geq 3$ (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 $|S|\geq 2|G|-1$ must either have every element of $G$ representable as a sum of $|G|$-terms from $S$ or else have all but $|G/H|-2$ of its terms lying in a common $H$-coset for some $H\leq G$. We show that the much weaker hypothesis $|S|\geq |G|+\exp(G)$ 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 $|S|$ is improved for several classes of groups $G$, yielding optimal lower bounds for $|S|$. We also generalize Olson's result for $|G|$-term subsums to an analogous one for $n$-term subsums when $n\geq \exp(G)$, 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.