A connection between covers of the integers and unit fractions

For integers a and n>0, let a(n) denote the residue class . Let A be a collection of finitely many residue classes such that A covers all the integers at least m times but 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 ∑s∈I1/ns−r/nk with I⊆{1,…,k−1}.