The $\{1,s\}$-weighted Davenport constant in $C_n^k$

Abstract

Let $G$ be a finite abelian group and let $\varnothing \ne A\subset \mathbb{Z}$. The $A$-weighted Davenport constant of $G$ is the smallest positive integer ${\mathsf{D}}_A(G)$ such that every sequence $x_1\cdot \dots \cdot x_{{\mathsf{D}}_A(G)}$ over $G$ has a non-empty subsequence $(x_{j_i}{)}_i$ such that ${\epsilon }_1{x}_{j_1}+{\epsilon }_2{x}_{j_2}+\dots +{\epsilon }_t{x}_{j_t}=0$ for some ${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_t\in A$. In this paper, we obtain both upper and lower bounds for ${\mathsf{D}}_{\{1,s\}}(C_n^k)$, where $C_n$ denotes the cyclic group of order $n$, $s^2\equiv 1(\mathrm{mod}n)$ and $s\not\equiv \pm 1(\mathrm{mod}n)$. These bounds become sharp in some "small" cases.

This is version 3 of the paper "The {1,s}-weighted Davenport constant in Z_n and an application to an inverse problem", whose version 2 was "The {1,s}-weighted Davenport constant in C_n^k and an application in an inverse problem". The weighted and the inverse problems were separeted, since we obtained a complete answer to the inverse problem without using the bounds given by the weighted problem

References in corpus (3)

• The Noether numbers and the Davenport constants of the groups of order less than 32
• An asymptotically tight bound for the Davenport constant
• Extremal product-one free sequences in $C_q{⋊}_s{C}_m$