Iterated sumsets and setpartitions

Abstract

Let $G\cong {\mathbb {Z}}/m_1{\mathbb {Z}}\times \cdots \times {\mathbb {Z}}/m_r{\mathbb {Z}}$ be a finite abelian group with $m_1\mid \cdots \mid m_r=\exp (G)$. 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 $n\ge \frac{1}{p}|G|-1$ or $n\ge \frac{1}{p}|G|+p-3$, 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 subsum version of Kneser’s Theorem was later shown to hold under the weaker hypothesis $n\ge {\mathsf {d}}^*(G)$, 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 $n\ge \exp (G)+1$ to obtain the same conclusions, with the bound further improved for several classes of near-cyclic groups.