Iterated Sumsets and Setpartitions

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 $n$-term subsums version of Kneser's Theorem, obtained either via the DeVos-Goddyn-Mohar Theorem or the Partition Theorem, has become a powerful tool used to prove numerous zero-sum and subsequence sum questions. It provides a structural description of sequences having a small number of $n$-term subsequence sums, ensuring this is only possible if most terms of the sequence are contained in a small number of $H$-cosets. For large $n\geq \frac1p|G|-1$ or $n\geq \frac1p|G|+p-3$, where $p$ is the smallest prime divisor of $|G|$, the structural description is particularly strong. In particular, most terms of the sequence become contained in a single $H$-coset, with additional properties holding regarding the representation of elements of $G$ as subsequence sums. This strengthened form of the subsums version of Kneser's Theorem was later to shown to hold under the weaker hypothesis $n\geq \mathsf d^*(G)$, where $\mathsf d^*(G)=\sum_{i=1}^{r}(m_i-1)$. In this paper, we reduce the restriction on $n$ even further to an optimal, best-possible value, showing we need only assume $n\geq \exp(G)+1$ to obtain the same conclusions, with the bound further improved for several classes of near-cyclic groups.