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

TL;DR

This paper provides a quantitative version of the Duffin-Schaeffer conjecture, showing that for almost all real numbers, the count of coprime solutions approximates a specific asymptotic formula, using advanced sieve and graph methods.

Contribution

It establishes a quantitative asymptotic count of coprime solutions for almost all reals, extending the Duffin-Schaeffer theorem with precise estimates.

Findings

Asymptotic count of solutions matches the sum of 2φ(q)ψ(q)/q for q up to Q

Method combines GCD graphs, sieve theory, and number-theoretic analysis

Demonstrates 'asymptotic independence' of approximation sets on average

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 coming 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.