迭代和集与集划分
Iterated Sumsets and Setpartitions
摘要：
设$G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$是一个有限阿贝尔群，其上有$m_1\mid\ldots\mid m_r=\exp(G)$ . Kneser定理的$n$ -term子和形式，通过DeVos-Goddyn-Mohar定理或分割定理得到，已经成为证明许多零和子序列和问题的有力工具。它提供了具有少量$n$ -项子序列和的序列的结构描述，确保这仅在序列的大多数项包含在少量$H$ -陪集中时才可能。对于大的$n\geq \frac1p|G|-1$或$n\geq \frac1p|G|+p-3$，其中$p$是$|G|$的最小素因子，结构描述特别强。特别地，序列的大多数项被包含在单个$H$陪集中，并且关于$G$的元素的表示作为子序列和的附加属性被保持。这种加强形式的子和版本的Kneser定理后来被证明持有较弱的假设$n\geq \mathsf d^*(G)$，其中$\mathsf d^*(G)=\sum_{i=1}^{r}(m_i-1)$。在本文中，我们进一步减少了对$n$的限制，使之成为最优的，最好的可能值，表明我们只需要假设$n\geq \exp(G)+1$就可以得到相同的结论，并且对几类近循环群的界进一步改进。
英文摘要：
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.
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