Rational approximations of irrational numbers

Abstract

Given quantities ${\Delta }_1,{\Delta }_2,\cdots ⩾0$, 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 $\phi$ for Euler’s totient function, we either have ${\sum }_{q=1}^{\infty }\phi (q){\Delta }_q=\infty$ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or ${\sum }_{q=1}^{\infty }\phi (q){\Delta }_q<\infty$ 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.