A weighted generalization of two theorems of Gao

作者：Grynkiewicz David J^*; Elimar Marchan Luz; Ordaz Oscar

来源：Ramanujan Journal, 2012, 28(3): 323-340.

DOI：10.1007/s11139-011-9350-x

• Zero-sum problem
• Davenport constant
• Weighted subsequence sums
• Setpartition

Let G be a finite abelian group and let AaS dagger a%26quot;currency sign 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)a %26lt;-s (r) such that for some w (i) aA. 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)a %26lt;-s (|G|), such that for some w (i) aA. In this paper, we show that %26lt;br%26gt;E-A(G) = vertical bar G vertical bar +D-A(G) - 1, %26lt;br%26gt;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 na parts per thousand yen|G|, in addition to the case n=|G|.

• 出版日期2012-8