ON m - COVERS AND m - SYSTEMS

Let A = {as(mod ns)}s=0 be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results related to the covering multiplicity of A. In particular, we show that if every integer lies in more than m0 = b ∑k s=1 1/nsc members of A, then for any a = 0, 1, 2, . . . there are at least ( m0 ba/n0c ) subsets I of {1, . . . , k} with ∑ s∈I 1/ns = a/n0. We also characterize when any integer lies in at most m members of A, where m is a fixed positive integer. 1. The main results For a ∈ Z and n ∈ Z = {1, 2, 3, . . . }, we simply denote the residue class {x ∈ Z : x ≡ a (mod n)} by a(n). For a finite system (1.1) A = {as(ns)}s=1 of residue classes, the function wA : Z → N = {0, 1, 2, . . . } given by (1.2) wA(x) = |{1 6 s 6 k : x ∈ as(ns)}| is called the covering function of A. Obviously wA(x) is periodic modulo the least common multiple N of the moduli n1, . . . , nk, and it is easy to see that the average ∑N−1 x=0 wA(x)/N equals ∑k s=1 1/ns. As in [S97] we call m(A) = minx∈Z wA(x) the covering multiplicity of system (1.1). Let m be any positive integer. If wA(x) > m for all x ∈ Z (i.e., m(A) > m), then (1.1) is said to be an m-cover of Z as in [S95, S96], and in this case ∑k s=1 1/ns > m. Covers (i.e. 1-covers) of Z were first introduced by P. Erdős [E50] and they are also called covering systems. If wA(x) = m for all x ∈ Z, then we call (1.1) an exact m-cover of Z as in [S96, S97] (and in this case ∑k s=1 1/ns = m). By [PZ, Theorem 1.3], when m > 2 there are exact m-covers of Z that cannot split into two covers of Z. If