Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911343403073536 |
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| author | Takahasi, Hiroki |
| author_facet | Takahasi, Hiroki |
| contents | The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal structure. We show that this set intersects the limit set of any parabolic iterated function system arising from the backward continued fraction in a set of full Hausdorff dimension. As a corollary, we show that the set of irrationals whose irrationality exponents are strictly bigger than $2$ and whose backward continued fraction expansions have bounded partial quotients is of Hausdorff dimension $1$. This is a sharp contrast to the fact that there exists no irrational whose irrationality exponent is strictly greater than $2$ and whose regular continued fraction expansion has bounded partial quotients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals Takahasi, Hiroki Number Theory Dynamical Systems The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal structure. We show that this set intersects the limit set of any parabolic iterated function system arising from the backward continued fraction in a set of full Hausdorff dimension. As a corollary, we show that the set of irrationals whose irrationality exponents are strictly bigger than $2$ and whose backward continued fraction expansions have bounded partial quotients is of Hausdorff dimension $1$. This is a sharp contrast to the fact that there exists no irrational whose irrationality exponent is strictly greater than $2$ and whose regular continued fraction expansion has bounded partial quotients. |
| title | Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2512.23402 |