On the metric theory of approximations by reduced fractions: Quantifying the Duffin-Schaeffer conjecture

  • Manuel Hauke (Speaker)

Activity: Talk or presentationTalk at conference or symposiumScience to science


Let $\psi: \mathbb{N} \to [0, 1/2]$ be given. Koukoulopoulos and Maynard (2020) proved the Duffin–Schaeffer conjecture: for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$
to the inequality $|\alpha−p/q| < \psi(q)/q$, if and only if the series
$\sum_{q=1}^{\infty} \varphi(q)\psi(q)/q$ is divergent. In
a recent joint work with Christoph Aistleitner and Bence Borda, we established a quantitative
version of this result in the following sense: for almost all $\alpha$, the number of coprime solutions $(p, q)$, subject to $q \leq Q$, is of asymptotic order $\psi(Q) = \sum_{q=1}^Q 2\varphi(q)\psi(q)/q$.
In this talk, I will give an overview of the original proof of Koukoulopoulos and Maynard and the additional ideas we used to obtain this quantification.
Period20 Jul 2022
Event title15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing: MCQMC 2022
Event typeConference
LocationLinz, AustriaShow on map
Degree of RecognitionInternational


  • Diophantine approximation