# Extreme Value Theory for Hurwitz Complex Continued Fractions

## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**.**Let $(\Omega ,\mathcal{B},\mu )$ denote a probability space and let ${\xi}_{j}:\Omega \to \mathbb{R}$ denote a stationary sequence of random variables. For natural numbers $u<v$, let ${\mathcal{M}}_{u,v}$ denote the smallest σ-algebra for which ${\xi}_{u},\dots ,{\xi}_{v}$ are measurable. Then, ${\xi}_{j}$ is said to be ψ-mixing if for any sets $A\in {\mathcal{M}}_{1,l}$ and $B\in {\mathcal{M}}_{l+n,\infty}$ we have:

#### 1.1. Extreme Value Theory for RCF Digits

**Theorem**

**1**

**.**Let ${M}_{n}={M}_{n}\left(x\right):={max}_{1\le i\le n}{a}_{n}\left(x\right)$. Then, for $r>0$ we have:

**Theorem**

**2**

**.**Let ${\xi}_{j}:\Omega \to \mathbb{R}$ denote a stationary and ψ-mixing sequence of random variables. For $\omega \in \Omega $ and $v\in \mathbb{R}$, set

#### 1.2. Extreme Value Theory for Other CF Algorithms

#### 1.3. Extreme Value Theory for Nearest Integer Continued Fractions

**Theorem**

**3.**

**Corollary**

**1.**

**Remark**

**1.**

**Theorem**

**4.**

**Proof.**

## 2. Complex Continued Fractions and Main Results

#### Hurwitz Complex Continued Fractions

**Theorem**

**5.**

**Corollary**

**2.**

## 3. The Invariant Measure with Respect to the Hurwitz Map

**Lemma**

**1**

#### Proof of Main Results

**Lemma**

**2.**

**Proof.**

**Remark**

**2.**

**Proof**

**of**

**Theorem 5.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Brezinski, C. History of Continued Fractions and Padé Approximants. In Computational Mathematics; Springer Series; Springer: Berlin, Germany, 1991; Volume 12. [Google Scholar]
- Iosifescu, M. Doeblin and the metric theory of continued fractions: A functional-theoretic solution to Gauss’ 1812 problem. In Doeblin and Modern Probability (Blaubeuren, 1991); Contemporary Mathematics—American Mathematical Society: Providence, RI, USA, 1993; Volume 149, pp. 97–110. [Google Scholar]
- Galambos, J. The distribution of the largest coefficient in continued fraction expansions. Quart. J. Math.
**1972**, 23, 147–151. [Google Scholar] [CrossRef] - Galambos, J. The largest coefficient in continued fractions and related problems. In Diophantine Approximation and Its Applications; Academic Press: New York, NY, USA, 1973; pp. 101–109. [Google Scholar]
- Iosifescu, M. A Poisson law for ψ-mixing sequences establishing the truth of a Doeblin’s statement. Rev. Roum. Math. Pures Appl.
**1977**, 22, 1441–1447. [Google Scholar] - Doeblin, W. Remarques sur la théorie métrique des fractions continues. Compos. Math.
**1940**, 7, 353–371. [Google Scholar] - Galambos, J. An iterated logarithm type theorem for the largest coefficient in continued fractions. Acta Arith.
**1973**, 25, 359–364. [Google Scholar] [CrossRef] [Green Version] - Philipp, W. A conjecture of Erdős on continued fractions. Acta Arith.
**1975**, 28, 379–386. [Google Scholar] [CrossRef] - Diamond, H.; Vaaler, J. Estimates for partial sums of continued fraction partial quotients. Pac. J. Math.
**1986**, 122, 73–82. [Google Scholar] [CrossRef] [Green Version] - Ghosh, A.; Kirsebom, M.; Roy, P. Continued fractions, the Chen-Stein method and extreme value theory. Ergod. Theory Dynam. Syst.
**2021**, 41, 461–470. [Google Scholar] [CrossRef] [Green Version] - Zweimüller, R. Hitting times and positions in rare events. arXiv
**2018**, arXiv:1810.10381v3. [Google Scholar] - Chang, J.; Chen, H. Slow increasing functions and the largest partial quotients in continued fraction expansions. Math. Proc. Camb. Philos. Soc.
**2018**, 164, 1–14. [Google Scholar] [CrossRef] - Chang, Y.; Ma, J. Some distribution results of the Oppenheim continued fractions. Monatsh. Math.
**2017**, 184, 379–399. [Google Scholar] [CrossRef] - Shen, L.; Xu, J.; Jing, H. On the largest degree of the partial quotients in continued fractions expansions over the field of formal Laurent series. Int. J. Number Theory
**2013**, 9, 1237–1247. [Google Scholar] [CrossRef] - Nakada, H.; Natsui, R. On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm. Monatsh Math
**2003**, 138, 267–288. [Google Scholar] [CrossRef] - González Robert, G. A complex Borel-Bernstein theorem. arXiv
**2021**, arXiv:2104.05129. [Google Scholar] - Schweiger, F. Ergodic Theory of Fibred Systems and Metric Number Theory; Oxford University Press: Oxford, UK, 1995. [Google Scholar]
- Waterman, M. Some ergodic properties of multi-dimensional F-expansions. Z Wahr Verw Geb.
**1970**, 16, 77–103. [Google Scholar] [CrossRef] [Green Version] - Schweiger, F. Multidimensional Continued Fractions; Oxford Science Publications; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Hurwitz, A. Über die Entwicklung complexer Grössen in Kettenbrüche (German). Acta Math.
**1887**, 11, 187–200. [Google Scholar] [CrossRef] - Rieger, G. Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen (German). J. Reine Angew. Math.
**1979**, 310, 171–181. [Google Scholar] - Rieger, G. Ein Gauss–Kusmin-Levy-Satz für Kettenbrüche nach nächsten Ganzen. (German). Manuscr. Math.
**1978**, 24, 437–448. [Google Scholar] [CrossRef] - Lukyanenko, A.; Vandehey, J. Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics. arXiv
**2018**, arXiv:1805.09312. [Google Scholar] - Hensley, D. Continued Fractions; World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, USA, 2006. [Google Scholar]
- Nakada, H. On the Kuzmin’s theorem for the complex continued fractions. Keio Eng. Rep.
**1976**, 29, 93–108. [Google Scholar] - Schweiger, F. Metrische Theorie einer Klasse zahlentheoretischer Transformationen (German). Acta Arith.
**1968**, 15, 1–18. [Google Scholar] [CrossRef] - Ei, H.; Ito, S.; Nakada, H.; Natsui, R. On the construction of the natural extension of the Hurwitz complex continued fraction map. Monatsh Math
**2019**, 188, 37–86. [Google Scholar] [CrossRef] - Schweiger, F. Kuzmin’s theorem revisited. Ergod. Theory Dyn. Syst.
**2000**, 20, 557–565. [Google Scholar] [CrossRef] - Schweiger, F. A new proof of Kuzmin’s theorem. Rev. Roum. Math. Pures Appl.
**2011**, 56, 229–234. [Google Scholar] - Hiary, G.; Vandehey, J. Calculations of the invariant measure for Hurwitz continued fractions. arXiv
**2019**, arXiv:1805.10151v2. [Google Scholar] [CrossRef] [Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kirsebom, M.S.
Extreme Value Theory for Hurwitz Complex Continued Fractions. *Entropy* **2021**, *23*, 840.
https://doi.org/10.3390/e23070840

**AMA Style**

Kirsebom MS.
Extreme Value Theory for Hurwitz Complex Continued Fractions. *Entropy*. 2021; 23(7):840.
https://doi.org/10.3390/e23070840

**Chicago/Turabian Style**

Kirsebom, Maxim Sølund.
2021. "Extreme Value Theory for Hurwitz Complex Continued Fractions" *Entropy* 23, no. 7: 840.
https://doi.org/10.3390/e23070840