Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912831998263296 |
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| author | Nakajima, Yuto Takahasi, Hiroki Wang, Baowei |
| author_facet | Nakajima, Yuto Takahasi, Hiroki Wang, Baowei |
| contents | Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions Nakajima, Yuto Takahasi, Hiroki Wang, Baowei Number Theory Dynamical Systems Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions. |
| title | Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2601.12737 |