Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases
type: Item
instance of: scholarly article
title: Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases
zbMATH Open document ID: 1448.11131
author: Jaap de Jonge
author: Cor Kraaikamp
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2018-08-30T00:00:00Z
review text: The authors consider the continued fraction convergents \(p_n/q_n\ (n\geq 1)\) of a real irrational number \(x\) in the form of a so called \textit{semi-regular continued fraction} (SRCF), denoted by \[\text{SRCF}:\quad [a_0;\varepsilon_1a_1,\varepsilon_2a_2,\varepsilon_3a_3,\ldots],\] where \(a_n\geq 1\ (n\geq 1),\varepsilon_{n+1}+a_n\geq 1\ (n\geq 1),\ \varepsilon_{n+1}+a_n\geq 2\) infinitely often. Specifically the \textit{approximation coefficients} are studied \[\theta_n(x)=q_n^ 2\Big|x-\frac{p_n}{q_n}\Big|\ (n\geq 1).\] Special types of these SRCF are the regular-, optimal- and nearest-integer continued fraction \[ \begin{matrix} \text{RCF}:& \quad \varepsilon_n=1,\ n\geq 1 \\ \text{OCF}:& \quad q_n\hbox{ maximal},\ \sup\theta_n\hbox{ minimal},\\ \text{NICF}:& \quad \varepsilon_n \in \{-1,1\}\ (n\geq 1),\ a_0\in\mathbb{Z},\ a_n\geq 2\ (n\geq 1),\ \varepsilon_{n+1}+a_n\geq 2\ (n\geq 1). \end{matrix} \] The aim is to calculate the asymptotic frequencies of the following six patterns: \[ \begin{matrix} \mathcal{A}:\ \theta_{n-1}<\theta_n<\theta_{n+1}, & \mathcal{B}:\ \theta_{n-1}<\theta_{n+1}<\theta_n, & \mathcal{C}:\ \theta_n<\theta_{n-1}<\theta_{n+1}, \\ \mathcal{D}:\ \theta_n<\theta_{n+1}<\theta_{n-1}, & \mathcal{E}:\ \theta_{n+1}<\theta_{n-1}<\theta_n, & \mathcal{F}:\ \theta_{n+1}<\theta_n<\theta_{n-1}. \end{matrix}\] This asymptotic frequency s defined by for instance \[AF(\mathcal{A})=\lim_{N\rightarrow\infty}\,\frac{1}{N}\#\{n\in\mathbb{N}\,|\,2\leq n\leq N,\ \theta_{n-1}<\theta_n<\theta_{n+1}\}.\] For the RCF these asymptotic values were given by \textit{H. Jager} and \textit{J. de Jonge} [Indag. Math., New Ser. 25, No. 4, 816--824 (2014; Zbl 1297.11083)]. In the paper under review, the authors give the explicit values for the OCF and the NICF: Theorem 4. For the optimal continued fraction, the asymptotic frequencies of the six patterns of three consecutive approximation constants are given by \[ AF_O(\mathcal{A})=AF_O(\mathcal{F})=\frac{3}{2}+\frac{1}{\l…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11K50
zbMATH DE Number: 6929336
zbMATH Keywords: continued fractions
zbMATH Keywords: metric theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1527904823
OpenAlex ID: W2805175355
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DOI: 10.2748/TMJ/1527904823
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description: scientific article; zbMATH DE number 6929336