On a different weighted zero-sum constant

TL;DR

This paper investigates a generalized zero-sum constant in finite abelian groups, focusing on weighted versions for specific subsets of integers, and determines exact values for certain weight-sets.

Contribution

It introduces and computes the weighted zero-sum constant $C_A(n)$ for particular subsets $A$, extending classical zero-sum theory to weighted and consecutive subsequences.

Findings

Determined $C_A(n)$ for specific weight-sets $A$.

Extended classical zero-sum results to weighted and consecutive subsequences.

Provided explicit formulas for certain cases.

Abstract

For a finite abelian group $(G,+)$, the constant $C(G)$ is defined to be the smallest natural number $k$ such that any sequence in $G$ having length $k$ will have a subsequence of consecutive terms whose sum is zero. For a subset $A\subseteq\mathbb Z_n$, the constant $C_A(n)$ is the smallest natural number $k$ such that any sequence in $G$ having length $k$ has an $A$-weighted zero-sum subsequence of consecutive terms. We determine the value of $C_A(n)$ for some particular weight-sets $A$.