Metrical properties of Hurwitz Continued Fractions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915159456350208 |
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| author | Bugeaud, Yann Robert, Gerardo Gonzalez Hussain, Mumtaz |
| author_facet | Bugeaud, Yann Robert, Gerardo Gonzalez Hussain, Mumtaz |
| contents | We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.%, paralleling the classical theory for regular continued fractions in real numbers.
Let $Φ:\mathbb{N}\to \mathbb{R}_{>0}$ be any function. For any complex number $z$ and $n\in\mathbb{N}$, let $a_n(z)$ denote the $n$th partial quotient in the Hurwitz continued fraction of $z$. One of the main results of this paper is the computation of the Hausdorff dimension of the set \[E(Φ) := \left\{ z\in \mathbb C: |a_n(z)|\geq Φ(n) \text{ for infinitely many }n\in\mathbb{N} \right\}. \] This study is a complex analog of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319--1339]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_08254 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Metrical properties of Hurwitz Continued Fractions Bugeaud, Yann Robert, Gerardo Gonzalez Hussain, Mumtaz Number Theory Complex Variables Dynamical Systems Metric Geometry We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued fractions, among other things, we determine that the space of valid sequences is not a closed set of sequences. Additionally, we establish a comprehensive metrical theory for Hurwitz continued fractions.%, paralleling the classical theory for regular continued fractions in real numbers. Let $Φ:\mathbb{N}\to \mathbb{R}_{>0}$ be any function. For any complex number $z$ and $n\in\mathbb{N}$, let $a_n(z)$ denote the $n$th partial quotient in the Hurwitz continued fraction of $z$. One of the main results of this paper is the computation of the Hausdorff dimension of the set \[E(Φ) := \left\{ z\in \mathbb C: |a_n(z)|\geq Φ(n) \text{ for infinitely many }n\in\mathbb{N} \right\}. \] This study is a complex analog of a well-known result of Wang and Wu [Adv. Math. 218 (2008), no. 5, 1319--1339]. |
| title | Metrical properties of Hurwitz Continued Fractions |
| topic | Number Theory Complex Variables Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2306.08254 |