Rational approximations of irrational numbers

TL;DR

This paper discusses the Duffin--Schaeffer conjecture in Diophantine approximation, highlighting its history, the conjecture's zero-one law, and the recent proof by Koukoulopoulos and Maynard that settled it.

Contribution

It reviews the history and key ideas of the Duffin--Schaeffer conjecture and presents the breakthrough proof by Koukoulopoulos and Maynard.

Findings

The Duffin--Schaeffer conjecture was proven true.

Almost all irrationals are approximable under certain divergence conditions.

The recent proof confirms the zero-one law in metric Diophantine approximation.

Abstract

Given quantities $\Delta_1,\Delta_2,\dots\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<\Delta_q$. Depending on the choice of $\Delta_q$ 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 $\varphi$ for Euler's totient function, we either have $\sum_{q=1}^\infty \varphi(q)\Delta_q=\infty$ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or $\sum_{q=1}^\infty\varphi(q)\Delta_q<\infty$ and almost no irrationals are approximable. We present the history of the Duffin--Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos--Maynard that settled it.