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
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Given quantities , a fundamental problem in Diophantine approximation is to understand which irrational numbers have infinitely many reduced rational approximations such that . Depending on the choice of and of , 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 for Euler’s totient function, we either have and then almost all irrational numbers (in the Lebesgue sense) are approximable, or 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.