Gao两个定理的加权推广
A Weighted Generalization of Two Theorems of Gao
摘要：
设$G$是有限阿贝尔群，$A\subseteq \mathbb{Z}$是非空的.令$D_A(G)$表示最小整数，使得$G$上任何长度为$D_A(G)$的序列必须包含非平凡子序列$s_1... s_r$，使得对于某些$w_i\in A$，$\sum_{i=1}^{r}w_is_i=0$。令$E_A(G)$表示最小整数，使得在$G$上长度为$E_A(G)$的任何序列必须包含长度为$|G|$，$s_1... s_{|G|}$的子序列，使得对于某些$w_i\in A$，$\sum_{i=1}^{|G|}w_is_i=0$。本文证明了$$E_A(G)=|G|+D_A(G)-1,$$证实了Thangadurai的一个猜想和Adhikari等人的期望，其中$A=\{1\}$是Gao的一个较早的结果，我们的结果推广了Adhikari，Rath，Chen，David，Urroz，Xia，Yuan，Zeng和Thangadurai所做的部分工作.此外，在适当的多重性限制下，我们证明了不仅可以用这种方式表示零，而且可以表示整个非平凡子群，并且如果这个子群不是全群$G$，则我们获得了序列的结构信息，推广了Gao的另一个非加权结果。除了$n=|G|$的情形外，我们的完整定理对更一般的具有$n\geq |G|$的$n$ -和也是有效的。
英文摘要：
Let $G$ be a finite abelian group and let $A\subseteq \mathbb{Z}$ be nonempty. Let $D_A(G)$ denote the minimal integer such that any sequence over $G$ of length $D_A(G)$ must contain a nontrivial subsequence $s_1... s_r$ such that $\sum_{i=1}^{r}w_is_i=0$ for some $w_i\in A$. Let $E_A(G)$ denote the minimal integer such that any sequence over $G$ of length $E_A(G)$ must contain a subsequence of length $|G|$, $s_1... s_{|G|}$, such that $\sum_{i=1}^{|G|}w_is_i=0$ for some $w_i\in A$. In this paper, we show that $$E_A(G)=|G|+D_A(G)-1,$$ confirming a conjecture of Thangadurai and the expectations of Adhikari, et al. The case $A=\{1\}$ is an older result of Gao, and our result extends much partial work done by Adhikari, Rath, Chen, David, Urroz, Xia, Yuan, Zeng and Thangadurai. Moreover, under a suitable multiplicity restriction, we show that not only can zero be represented in this manner, but an entire nontrivial subgroup, and if this subgroup is not the full group $G$, we obtain structural information for the sequence generalizing another non-weighted result of Gao. Our full theorem is valid for more general $n$-sums with $n\geq |G|$, in addition to the case $n=|G|$.
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