On the densities of rational multiples

Abstract

For two subsets of natural numbers $ A,B \subset \mathbb{N} $ , define the set of rational numbers $ \mathcal{M}\left( {A,B} \right) $ with the elements represented by m/n, where m and n are coprime, m is divisible by some a ∈ A, and n is divisible by some b ∈ B. Let I be some interval of positive real numbers and $ \mathcal{F}_x^I $ denote the set of rational numbers m/n ∈ I such that m and n are coprime and n ⩽ x. The analogue to the Erdös–Davenport theorem about multiples is proved: under some constraints on I, the limits $ {{{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I \cap \mathcal{M}\left( {A,B} \right)} \right\}} }} \left/ {{\sum {\left\{ {\frac{1}{{mn}}:\frac{m}{n} \in \mathcal{F}_x^I} \right\}} }} \right.} $ exist for all subsets $ A,B \subset \mathbb{N} $ as x → ∞.