On the Function w(x)=|{1≤s≤k : x≡a _s (mod n _s )}|

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 ≡ a _s (mod n _s )}|. 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 n ₀ of w(x) and the moduli n ₁, . . . , n _k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n _k /(n ₀,n _k )−1 there exists a J c{1, . . . , k−1} such that ${\sum\nolimits_{s \in J} {1/n_{s} = r/n_{k} } }$. (b) If n ₁ ≤···≤n _k−l <n _k−l+1 =···=n _k (0 < l < k), then for any positive integer r < n _k /n _k−l with r ≢ 0 (mod n _k /(n ₀,n _k )), 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 n _k . (c) max_(x∈ℤ w(x) can be written in the form ${\sum\nolimits_{{\left( {s = 1} \right)}}^k {m_{s} /n_{s} } }$ where m ₁, . . .,m _k are positive integers.