Generalization of some weighted zero-sum theorems and related Extremal sequence

Abstract: Let $G$ be a finite abelian group of exponent $n$ and let $A$ be a non-empty subset of $[1,n-1]$. The Davenport constant of $G$ with weight $A$, denoted by $D_A(G)$, is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has a non-empty $A$-weighted zero-sum subsequence. Similarly, the combinatorial invariant $E_{A}(G)$ is defined to be the least positive integer $\ell$ such that any sequence over $G$ of length $\ell$ has an $A$-weighted zero-sum subsequence of length $|G|$. In this article, we determine the exact value of $D_A(\mathbb{Z}_n)$, for some particular values of $n$, where $A$ is the set of all cubes in $\mathbb{Z}_n^*$. We also determine the structure of the related extremal sequence in this case.