Davenport constant with weights

TL;DR

This paper introduces weighted Davenport constants for cyclic groups, establishing a key relationship between two such constants and solving a problem posed by multiple researchers in additive combinatorics.

Contribution

The paper defines weighted Davenport constants for cyclic groups and proves that the minimal sequence length ensuring zero-sum subsequences relates directly to the weighted Davenport constant, resolving an open problem.

Findings

Proved that $E_A(n)=D_A(n)+n-1$ for weighted Davenport constants.

Established the relationship between $E_A(n)$ and $D_A(n)$ in weighted zero-sum problems.

Solved an open problem posed by Adhikari, Rath, Chen, Thangadurai, and Griffiths.

Abstract

For the cyclic group $G=\mathbb{Z}/n\mathbb{Z}$ and any non-empty $A\in\mathbb{Z}$. 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 $\sum_{i=1}^l a_ix_{j_i} = 0$. Similarly, we define the constant $E_A(n)$ to be the least $t\in\mathbb{N}$ such that for all sequences $(x_1, >..., x_t)$ with $x_i \in G$, there exist indices $j_1, ..., j_n\in\mathbb{N}, 1\leq j_1 <... < j_n\leq t$, and $\vartheta_1, >..., \vartheta_n\in A$ with $\sum^{n}_{i=1} \vartheta_ix_{j_i} = 0$. 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}.