# Rational approximations of irrational numbers

### Dimitris Koukoulopoulos

Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada

This book chapter is published *open access.*

## Abstract

Given quantities $Δ_{1},Δ_{2},⋯⩾0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $∣x−a/q∣<Δ_{q}$. Depending on the choice of $Δ_{q}$ and of $x$, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a “metric” point of view, the question is governed by a simple zero–one law: writing $φ$ for Euler’s totient function, we either have $∑_{q=1}φ(q)Δ_{q}=∞$ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or $∑_{q=1}φ(q)Δ_{q}<∞$ and almost no irrationals are approximable. We will present the history of the Duffin–Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos–Maynard that settled it.