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arXiv · 2504.16565

General Duffin--Schaeffer-type counterexamples in diophantine approximation

Sam Chow·Manuel Hauke·Andrew Pollington·Felipe A. Ramírez
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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 Erd\H{o}s, Vaaler, and Yu.

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Sam Chow, Manuel Hauke, Andrew Pollington, Felipe A. Ramírez. 2025-04-23. General Duffin--Schaeffer-type counterexamples in diophantine approximation. https://arxiv.org/abs/2504.16565

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