Remarks on the plus-minus weighted Davenport constant

TL;DR

This paper investigates the plus-minus weighted Davenport constant for finite abelian groups, providing new bounds and exact values for specific group types, advancing understanding of weighted zero-sum problems.

Contribution

It introduces new bounds, especially lower bounds, and determines the exact values of the constant for various classes of finite abelian groups.

Findings

New lower bounds for the plus-minus weighted Davenport constant.

Exact values computed for specific types of finite abelian groups.

Enhanced understanding of weighted zero-sum sequences in group theory.

Abstract

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsf{D}_{\pm}(G)$, is the smallest $\ell$ such that each sequence $g_1 ... g_{\ell}$ over $G$ has a weighted zero-subsum with weights +1 and -1, i.e., there is a non-empty subset $I \subset \{1,..., \ell\}$ such that $\sum_{i \in I} a_i g_i =0$ for $a_i \in \{+1,-1\}$. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups.