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

Part of: Elementary number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  03 February 2023

Christoph Aistleitner ,
Bence Borda  and
Manuel Hauke
Christoph Aistleitner
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II, 8010 Graz, Austria aistleitner@math.tugraz.at
Bence Borda
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II, 8010 Graz, Austria borda@math.tugraz.at
Manuel Hauke
Affiliation:
Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II, 8010 Graz, Austria hauke@math.tugraz.at
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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.

Keywords

Diophantine approximationmetric number theoryDuffin–Schaeffer conjectureKoukoulopoulos– Maynard theorem

MSC classification

Primary: 11J83: Metric theory
Secondary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors 11J04: Homogeneous approximation to one number 11K60: Diophantine approximation
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 2 , February 2023 , pp. 207 - 231
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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