Iterated Sumsets and Setpartitions

TL;DR

This paper improves the structural understanding of zero-sum sequences in finite abelian groups by establishing optimal bounds for the number of terms needed to guarantee certain sumset properties, refining previous results.

Contribution

It proves that the structural description holds for the smallest possible n, specifically n ≥ exp(G)+1, enhancing the known bounds for various classes of groups.

Findings

Structural description holds for n ≥ exp(G)+1

Bound is optimal and best-possible

Improved bounds for near-cyclic 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 $n$-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.