General Duffin--Schaeffer-type counterexamples in diophantine approximation

2025/04/23 by Sam Chow, Chow, Sam, Manuel Hauke +5 · 1 citation
Mathematics · Physics and Astronomy · #11J83 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2504.16565

openalex publication_date 2025/04/23 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erdős, Vaaler, and Yu.

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