A sharp result on $m$-covers

Let $A=\{a_{s}+n_{s}\mathbb{Z} \}_{s=1}^{k}$ be a finite system of residue classes which forms an $m$-cover of $\mathbb{Z}$ (i.e., every integer belongs to at least $m$ members of $A$). In this paper we show the following sharp result: For any positive integers $m_{1},\ldots ,m_{k}$ and $\theta \in [0,1)$, if there is $I\se \{1,\ldots ,k\}$ such that the fractional part of $\sum _{s\in I} m_{s}/n_{s}$ is $\theta$, then there are at least $2^{m}$ such subsets of $\{1,\ldots ,k\}$. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to $m$-covers of the integral ring of any algebraic number field with a power integral basis.