A Theorem concerning Systems of Residue Classes

By system we mean a multi-set whose elements are unordered but may occur repeatedly. FollowingŠ. Znám [8] we use a(n) to denote the residue class { x ∈ Z : x ≡ a (mod n) }. For a system (1) A = {a i (n i)} k i=1 of residue classes, the n i are called its moduli. Definition. An integer T is said to be a covering period of (1) if it is a period of the characteristic function of the set k i=1 a i (n i). It is clear that [n 1 ,. .. , n k ] is a covering period of (1), and that any covering period is a multiple of the smallest positive one. For any set S of integers we use d(S) to denote the asymptotic density lim N →∞