On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos–Maynard theorem

Abstract

Let$\psi : \mathbb {N} \to [0,1/2]$be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals$\alpha$there are infinitely many coprime solutions$(p,q)$to the inequality$|\alpha - p/q| < \psi (q)/q$, provided that the series$\sum _{q=1}^\infty \varphi (q) \psi (q) / q$is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all$\alpha$the number of coprime solutions$(p,q)$, subject to$q \leq Q$, is of asymptotic order$\sum _{q=1}^Q 2 \varphi (q) \psi (q) / q$. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate from sieve theory, and number-theoretic input on the ‘anatomy of integers’. The key phenomenon is that the system of approximation sets exhibits ‘asymptotic independence on average’ as the total mass of the set system increases.