Weighted zero-sum problems over $C_3^r$

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 $41\leq s_A(C_{3}^{5})\leq45$.