Rational approximations of irrational numbers

2021/09/22 by Dimitris Koukoulopoulos, Koukoulopoulos, Dimitris
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.11003

openalex publication_date 2021/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given quantities Δ₁,Δ₂,…\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|<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 present the history of the Duffin--Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos--Maynard that settled it.

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