Extremal sequences for a weighted zero-sum constant

TL;DR

This paper investigates the extremal sequences in cyclic groups that avoid weighted zero-sum subsequences, providing characterizations for specific weight sets to understand their structure.

Contribution

It characterizes extremal sequences for the weighted zero-sum constant in cyclic groups for particular weight sets, advancing understanding of zero-sum problems.

Findings

Characterization of extremal sequences for certain weight sets

Identification of conditions preventing weighted zero-sum subsequences

Extension of zero-sum theory to weighted and consecutive subsequences

Abstract

The constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of consecutive terms whose $A$-weighted sum is zero, where the weight set $A\subseteq \mathbb Z_n\setminus \{0\}$. If $C_A(n)=k$, then a sequence in $\mathbb Z_n$ of length $k-1$ which has no $A$-weighted zero-sum subsequence of consecutive terms is called an $A$-extremal sequence. We characterize these sequences for some particular weight sets.