Davenport constant with weights

Pingzhi Yuan; Xiangneng Zeng

paper

Davenport constant with weights

arXiv:0909.2388

Abstract

For the cyclic group $G = Z / n Z$ and any non-empty $A \in Z$ . We define the Davenport constant of $G$ with weight $A$ , denoted by $D_{A} (n)$ , to be the least natural number $k$ such that for any sequence $(x_{1}, ..., x_{k})$ with $x_{i} \in G$ , there exists a non-empty subsequence $(x_{j_{1}}, ..., x_{j_{l}})$ and $a_{1}, ..., a_{l} \in A$ such that $\sum_{i = 1}^{l} a_{i} x_{j_{i}} = 0$ . Similarly, we define the constant $E_{A} (n)$ to be the least $t \in N$ such that for all sequences $(x_{1}, > ..., x_{t})$ with $x_{i} \in G$ , there exist indices $j_{1}, ..., j_{n} \in N, 1 \leq j_{1} < ... < j_{n} \leq t$ , and $ϑ_{1}, > ..., ϑ_{n} \in A$ with $\sum_{i = 1}^{n} ϑ_{i} x_{j_{i}} = 0$ . In the present paper, we show that $E_{A} (n) = D_{A} (n) + n - 1$ . This solve the problem raised by Adhikari and Rath \cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and Griffiths \cite{gr08}.

6pages