A connection between covers of the integers and unit fractions
Zhi-Wei Sun
Abstract
For integers a and n>0, let a(n) denote the residue class x∈ Z: x=a (mod n). Let A be a collection as(ns)s=1k of finitely many residue classes such that A covers all the integers at least m times but as(ns)s=1k-1 does not. We show that if nk is a period of the covering function wA(x)=|1 s k: x∈ as(ns)| then for any r=0,...,nk-1 there are at least m integers in the form Σs∈ I1/ns-r/nk with I contained in 1,...,k-1.
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