Duffin--Schaeffer conjecture for coprime rational approximations

Let $\Phi:\mathbb N\to\mathbb R_{\geqslant 0}$ satisfy the hypothesis denoted by in the source. Duffin--Schaeffer conjecture. For almost all ${\beta}\in\mathbb R$, the inequality

$$|n{\beta}-r|<\Phi(n)$$

holds for infinitely many coprime pairs $(n,r)\in\mathbb N\times\mathbb Z$.

This is the homogeneous Duffin--Schaeffer formulation underlying the paper's discussion of metric rational approximation; the supplied excerpt does not identify the full content of or provide resolution evidence.