A Connection between Covers of the Integers and Unit Fractions
نویسندگان
چکیده
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=1 of finitely many residue classes such that A covers all the integers at least m times but {as(ns)} s=1 does not. We show that if nk is a period of the covering function wA(x) = |{1 6 s 6 k : x ∈ as(ns)}| then for any r = 0, . . . , nk−1 there are at least m integers in the form ∑ s∈I 1/ns − r/nk with I ⊆ {1, . . . , k − 1}. MSC: primary 11B25; secondary 11B75, 11D68.
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