Zaremba's Conjecture for Geometric Sequences: An Algorithm
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914281732177920 |
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| author | Dubno, Elias |
| author_facet | Dubno, Elias |
| contents | Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11279 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zaremba's Conjecture for Geometric Sequences: An Algorithm Dubno, Elias Number Theory 11A55, 11J70 Even though Zaremba's conjecture remains open, Bourgain and Kontorovich solved the problem for a full density subset. Nevertheless, there are only a handful of explicit sequences known to satisfy the strong version of the conjecture, all of which were obtained using essentially the same algorithm. In this note, we provide a refined algorithm using the folding lemma for continued fractions, which both generalizes and improves on the old one. As a result, we uncover new examples that fulfill the strong version of Zaremba's conjecture. |
| title | Zaremba's Conjecture for Geometric Sequences: An Algorithm |
| topic | Number Theory 11A55, 11J70 |
| url | https://arxiv.org/abs/2310.11279 |