The Weighted Davenport Constant of a group and a related extremal problem

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp (G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary $G$-sequence $(x_1,\dots ,x_k)$, there exists a non-empty subsequence $(x_{i_1},\dots ,x_{i_t})$ along with $a_{j}\in A$suchthat$\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$: For an integer $k\ge 2$, determine ${\text{fD}}_G(k):=\min \{\mathrm{\mid }A\mathrm{\mid }:D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ `not-too-small’, this is a well-posed problem and one of the most interesting cases occurs for $G={\mathbb{Z}}_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.