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

Description

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.

Period	20 Jul 2022
Event title	15th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing: MCQMC 2022
Event type	Conference
Location	Linz, AustriaShow on map
Degree of Recognition	International