迭代和集与子序列和
Iterated Sumsets and Subsequence Sums
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$m_1\mid\ldots\mid m_r=\exp(G)$ . Kemperman结构定理刻画了满足$|A+B|<|A|+|B|$的所有子集$A,\,B\subseteq G$，并已被推广到覆盖当$|A+B|\leq |A|+|B|$。利用这些结果，我们提供了一个精确的结构描述的所有有限子集$A\subseteq G$与$|nA|\leq (|A|+1)n-3$时$n\geq 3$（也当$G$是无限的），在这种情况下，许多病理可能性的情况下$n=2$消失，特别是大$n\geq \exp(G)-1$。结构描述与其他参数相结合，以推广奥尔森断言的子序列和结果，即具有长度$|S|\geq 2|G|-1$的来自$G$的项的序列$S$必须具有可表示为来自$S$的$|G|$ -项的和的$G$的每个元素，否则具有除了$|G/H|-2$之外的所有项，其项位于某些$H\leq G$的公共$H$ -陪集中。我们表明，弱得多的假设$|S|\geq |G|+\exp(G)$足以获得一个几乎相同的结论，在$H$是平凡的情况下，我们必须允许所有的$S$，但$|G/H|-1$条款是从相同的$H$陪集。对PDMATHBNF上的界进行了改进，得到了PDMATHBNF的最优下界。我们还推广奥尔森的结果$|G|$ -长期子和类似的$n$ -长期子和时，$n\geq \exp(G)$，与界同样改善了几个特殊类的群体。这改进了Olson的结果的先前推广，具有$n$最优的界限。
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 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.
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