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