On the densities of rational multiples

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 → ∞.