Generalization of some weighted zero-sum theorems and related Extremal sequence

TL;DR

This paper extends zero-sum theorems to weighted cases in finite abelian groups, specifically determining exact values of weighted Davenport constants for certain groups and subsets, and analyzing extremal sequences.

Contribution

It generalizes weighted zero-sum theorems by calculating exact Davenport constants for specific groups and weights, and characterizes extremal sequences.

Findings

Exact value of $D_A(Z_n)$ for particular $n$ and $A$

Structure of extremal sequences in these cases

Extension of zero-sum theorems to weighted settings

Abstract

Let $G$ be a finite abelian group of exponent $n$ and let $A$ be a non-empty subset of $[1,n-1]$. The Davenport constant of $G$ with weight $A$, denoted by $D_A(G)$, is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has a non-empty $A$-weighted zero-sum subsequence. Similarly, the combinatorial invariant $E_{A}(G)$ is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has an $A$-weighted zero-sum subsequence of length $|G|$. In this article, we determine the exact value of $D_A(\mathbb{Z}_n)$, for some particular values of $n$, where $A$ is the set of all cubes in $\mathbb{Z}_n^*$. We also determine the structure of the related extremal sequence in this case.