A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard

2025/05/13 by Manuel Hauke, Hauke, Manuel · 1 citation
Mathematics · Physics and Astronomy · #11J83 11K60 11J04 11A05 #Advanced Mathematical Theories and Applications #Conjecture #Diophantine approximation #Diophantine equation #FOS: Mathematics #Focus (optics) #Foundation (evidence) #History and Theory of Mathematics #Key (lock) #Mathematical Dynamics and Fractals #Metric (unit) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2505.08901

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this note, we review the history of Khintchine's Theorem which is the foundation of metric Diophantine approximation, and discuss several generalizations and recent breakthroughs in this area. We focus particularly on the direction of the Duffin-Schaeffer Conjecture which was spectacularly proven in 2020. We present some simplified key ideas of the proof that can also be applied in various other areas of number theory.

Citations

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Cited by

• Hitting times of shrinking targets: Transversality and an ergodic theorem

Related

• Classical metric Diophantine approximation revisited
• Variants of Khintchine's theorem in metric Diophantine approximation
• Proving the Duffin-Schaeffer conjecture without GCD graphs
• Metric Diophantine Approximation: Aspects of Recent Work
• Metric Diophantine Approximation: aspects of recent work