Weighted Zero-Sum Problems Over $C_3^r$

TL;DR

This paper investigates the minimal sequence length in the group $C_3^r$ needed to guarantee an zero-sum subsequence with elements from $\

Contribution

It provides new estimates and exact values for the weighted zero-sum problem over groups of the form $C_3^r$, advancing understanding of zero-sum sequences.

Findings

Exact value: $s_A(C_3^3)=9$

Exact value: $s_A(C_3^4)=21$

Bounds: $41 \,\leq s_A(C_3^5)\leq45$

Abstract

Let $C_n$ be the cyclic group of order $n$ and set $s_{A}(C_n^r)$ as the smallest integer $\ell$ such that every sequence $\mathcal{S}$ in $C_n^r$ of length at least $\ell$ has an $A$-zero-sum subsequence of length equal to $\exp(C_n^r)$, for $A=\{-1,1\}$. In this paper, among other things, we give estimates for $s_A(C_3^r)$, and prove that $s_A(C_{3}^{3})=9$, $s_A(C_{3}^{4})=21$ and $41\leq s_A(C_{3}^{5})\leq45$.