General Duffin--Schaeffer-type counterexamples in diophantine approximation
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916703558959104 |
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| author | Chow, Sam Hauke, Manuel Pollington, Andrew Ramírez, Felipe A. |
| author_facet | Chow, Sam Hauke, Manuel Pollington, Andrew Ramírez, Felipe A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | General Duffin--Schaeffer-type counterexamples in diophantine approximation Chow, Sam Hauke, Manuel Pollington, Andrew Ramírez, Felipe A. Number Theory 11J83 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. |
| title | General Duffin--Schaeffer-type counterexamples in diophantine approximation |
| topic | Number Theory 11J83 |
| url | https://arxiv.org/abs/2504.16565 |