On Covering Multiplicity

Let A = {as +nsZ}s=1 be a system of arithmetic sequences which forms an m-cover of Z (i.e. every integer belongs at least to m members of A). In this paper we show the following surprising properties of A: (a) For each J ⊆ {1, · · · , k} there exist at least m subsets I of {1, · · · , k} with I 6= J such that ∑ s∈I 1/ns − ∑ s∈J 1/ns ∈ Z. (b) If A forms a minimal m-cover of Z, then for any t = 1, · · · , k there is an αt ∈ [0, 1) such that for every r = 0, 1, · · · , nt − 1 there exists an I ⊆ {1, · · · , k} \ {t} for which [ ∑ s∈I 1/ns] > m− 1 and { ∑ s∈I 1/ns} = (αt + r)/nt.