关于加权零和序列
On weighted zero-sum sequences
摘要：
设G是指数为exp（G）=n>1的有限可加阿贝尔群，A是{1，…，n-1}。本文研究了最小正整数$m$，记为s_A（G），使得任意序列{c_i}_{i=1}^m都有一个长度n=exp（G）的子序列{ c_{i_j} }_{j=1}^n，其中有a_1，. a_n，使得sum_{j=1} ^na_ ic_{i_j} =0。当G是p-群，A不含p的倍数，且A的任意两个不同元素模p不全等时，证明了s_A（G）至多为$\lceil D(G)/|A|\rceil+exp(G)-1$，如果|A|至少为（D（G）-1）/（exp（G）-1），其中D（G）是G的Davenport常数，且s_A（G）关于|A|的上界本质上是最好的.在A={1，-1}的情形下，我们确定了当exp（G）为偶数时s_{1,-1}（G）的渐近性态，证明了对于偶数指数固定秩的有限阿贝尔群，当exp（G）趋于无穷大时，s_{1,-1}（G）=exp（G）+log_2 |G| +O（log_2log_2 |G|）.结合$exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$的下限，其中$G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$，1<n_1|... |n_r，这确定了s_{1,-1}（G），对于偶数指数组，直到小阶误差项。我们的方法利用了L-交集系统的理论。还计算了与s_{1,-1}（G）有关的一些额外的更具体的值和结果。
英文摘要：
Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{1,-1}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{1,-1}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{1,-1}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{1,-1}(G) are also computed.
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