A Weighted Generalization of Two Theorems of Gao
| dc.creator | Grynkiewicz, David J. | |
| dc.creator | Marchan, Luz Elimar | |
| dc.creator | Ordaz, Oscar | |
| dc.date | 2009-03-16 | |
| dc.date.accessioned | 2026-07-07T12:52:55Z | |
| dc.date.available | 2026-07-07T12:52:55Z | |
| dc.description | 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 $n=|G|$. | |
| dc.identifier | https://arxiv.org/abs/0903.2810 | |
| dc.identifier | http://arxiv.org/abs/0903.2810 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/223453 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11B75, 20K01 | |
| dc.title | A Weighted Generalization of Two Theorems of Gao | |
| dc.type | text |