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

For a finite system A = {as + nsZ}s=1 of arithmetic sequences the covering function is w(x) = |{1 6 s 6 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 J ⊆ {1, · · · , k−1} such that ∑ s∈J 1/ns = r/nk. (b) If n1 6 · · · 6 nk−l < nk−l+1 = · · · = nk (0 < l < k), then for any positive integer r < nk/nk−l with r 6≡ 0 (mod nk/(n0, nk)), the binomial coefficient (l r ) can be written as the sum of some (not necessarily distinct) prime divisors of nk. (c) maxx∈Z w(x) can be written in the form ∑k s=1ms/ns where m1, · · · ,mk are positive integers. 2000 Mathematics Subject Classifications. Primary 11B25; Secondary 05A15, 11A07, 11A25, 11B75, 11D68.