Weighted Zero-Sum Problems Over $C_3^r$

Let $C_n$ be the cyclic group of order $n$ and set $s_A(C_n^r)$ as the smallest integer $\mathrm{\ell }$ such that every sequence $\mathcal{S}$ in $C_n^r$ of length at least $\mathrm{\ell }$ 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 $41\le s_A(C_3^5)\le 45$.