On the application of dependence with complete connections to the metrical theory of G-continued fractions. Dependence with complete connections
type: Item
instance of: scholarly article
title: On the application of dependence with complete connections to the metrical theory of G-continued fractions. Dependence with complete connections
zbMATH Open document ID: 0644.10035
DOI: 10.1007/BF00972019
author: B. George
published in: Lithuanian Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Every irrational x in the interval [G-2, G], with \(G=(1+\sqrt{5})/2\), has a continued fraction expansion of the form \(x=\epsilon_ 1/(\alpha_ 1+\epsilon_ 2/(\alpha_ 2+...\), where \(\epsilon_ j\) is either -1 or \(+1\), and each digit \(\alpha_ j\) is an odd positive integer. Via the theory of random systems with complete connections the author establishes a number of deep results on the digits \(\alpha_ j\), \(j\geq 1\), and on the approximants \(p_ n/q_ n=\epsilon_ 1/(\alpha_ 1+\epsilon_ 2/(\alpha_ 2+...+\epsilon_ n/\alpha_ n)...)\) including the limit relation, as \(n\to +\infty\), \(\lim n^{-1} \log | x-p_ n/q_ n| =-\pi^ 2/9 \log G\) almost everywhere.
Mathematics Subject Classification ID: 11K16
zbMATH DE Number: 4049734
zbMATH Keywords: G-continued fraction
zbMATH Keywords: uniformly ergodic
zbMATH Keywords: strong laws of large number
zbMATH Keywords: approximations
zbMATH Keywords: entropy
zbMATH Keywords: random systems with complete connections
zbMATH Keywords: digits
zbMATH Keywords: approximants
reviewed by: Janos Galambos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4049734