$Q_p$-weighted zero-sum constants

A sequence $S=(x_1,\dots ,x_k)$ in ${\mathbb{Z}}_p$ is called a $(Q_p,\mathbf{1})$-weighted zero-sum sequence if there exist $a_1,\dots ,a_k\in Q_p$ such that $a_1{x}_1+\cdots +a_k{x}_k=0$ and $a_1+\cdots +a_k=0$. The constant $E_{Q_p,\mathbf{1}}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in ${\mathbb{Z}}_p$ has a $(Q_p,\mathbf{1})$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf{1}}$ and the related constants $C_{Q_p,\mathbf{1}}$ and $D_{Q_p,\mathbf{1}}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$.