  Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which
forms an m-cover of Z (i.e., every integer belongs at least to m members of A).
In this paper we show the following sharp result: For any positive integers
m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,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,...,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.
