On the Function w(x)=|{1≤s≤k : x≡as (mod ns)}|

For a finite system $$ A = {\left\{ {a_{s} + n_{s} \mathbb{Z}} \right\}}^{k}_{{s = 1}} $$ of arithmetic sequences the covering function is w(x) = |{1 ≤ s ≤ k : x ≡ as (mod ns)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n0 of w(x) and the moduli n1, . . . , nk in such a system A. Here are three central results: (a) For each r=0,1, . . .,nk/(n0,nk)−1 there exists a Jc{1, . . . , k−1} such that $$ {\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } } $$. (b) If n1 ≤···≤nk−l <nk−l+1 =···=nk (0 < l < k), then for any positive integer r < nk/nk−l with r ≢ 0 (mod nk/(n0,nk)), the binomial coefficient $$ {\left( {\begin{array}{*{20}c} {l} \\ {r} \\ \end{array} } \right)} $$ can be written as the sum of some (not necessarily distinct) prime divisors of nk. (c) max(x∈ℤw(x) can be written in the form $$ {\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } } $$ where m1, . . .,mk are positive integers.