Christophe Aistleitner, Graz
Title: On the metric theory of approximations by reduced fractions.
Abstract: In 2019 Dimitris Koukoulopoulos and James Maynard solved the Duffin-Schaeffer conjecture, a central problem in metric Diophantine approximation that had been open since 1941. Very roughly speaking, the Koukoulopoulos-Maynard theorem states that there is a simple convergence/divergence criterion which allows to decide whether (Lebesgue-)almost all real numbers allow infinitely many coprime rational approximations of a certain quality, or not. In this talk I will report on very recent joint work with Bence Borda and Manuel Hauke (both from TU Graz as well) which goes beyond the existence of infinititely many solutions, and gives an actual asymptotics for the typical number of coprime rational approximations up to a certain threshold in the divergence case. I will relate some of the history of the subject, and try to convey some of the (probablistic) philosophy behind the problem. The proof relies mainly on sieve theory and the "anatomy of integers", and in particular on the method of GCD graphs which was introduced by Koukoulopoulos-Maynard in their proof.
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