How many points contain homothetic copies in their Hurwitz continued fraction expansion?
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914278109347840 |
|---|---|
| author | Nakajima, Yuto Takahasi, Hiroki |
| author_facet | Nakajima, Yuto Takahasi, Hiroki |
| contents | We prove that the set of complex irrationals whose partial quotients in their Hurwitz continued fraction expansion are naturally regarded as subsets of $\mathbb Z^2$ and contain infinitely many homothetic copies of any finite subset of $\mathbb Z^2$ is of Hausdorff dimension $1$. Our result provides a clear and concrete example of multidimensional pattern emergence in number-theoretic expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How many points contain homothetic copies in their Hurwitz continued fraction expansion? Nakajima, Yuto Takahasi, Hiroki Dynamical Systems Number Theory We prove that the set of complex irrationals whose partial quotients in their Hurwitz continued fraction expansion are naturally regarded as subsets of $\mathbb Z^2$ and contain infinitely many homothetic copies of any finite subset of $\mathbb Z^2$ is of Hausdorff dimension $1$. Our result provides a clear and concrete example of multidimensional pattern emergence in number-theoretic expansions. |
| title | How many points contain homothetic copies in their Hurwitz continued fraction expansion? |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2601.17854 |