Recurrence speed of multiples of an irrational number.
type: Item
instance of: scholarly article
title: Recurrence speed of multiples of an irrational number.
zbMATH Open document ID: 1040.37004
author: Geon Ho Choe
author: Byoung Ki Seo
published in: Proceedings of the Japan Academy. Series A
publication date: 2002-09-12T00:00:00Z
review text: Let \(\theta\) be an irrational number in \((0,1)\) and \(x\) a real number in \((0,1)\). Define the transformation of \(x\) by \(T_\theta(x)= \theta+ x\); for its iteration we have \(T^n_\theta(x)= n\theta+ x\). For a real \(t\) set \(\| t\|= \min(t- n)\), where the minimum has to be taken over all integer \(n\). Denote \(Q_n\) the interval \(((i- 1)/2^n\), \(i/2^n)\) containing \(x\). Set \[ J_n(x)= \min(j\geq 1:\| x- T^j_\theta x\|< 1/2^n), \] and \[ K_n(x)= \min(j\geq 1: T^j_\theta(x)\in Q_n). \] The authors prove that \[ \log J_n(x)/n\to 1\quad\text{and}\quad \log K_n(x)\to 1,\tag{\(*\)} \] almost every in \(\theta\) as \(n\to\infty\). Reviewer's remark: The two asymptotic relations in \((*)\) are equivalent. The authors are almost immediate consequences of a classical result of Khitchine, stating that if \(A_n/B_n\) are the convergents of the continued fraction expansion of \(\theta\), then we have almost everywhere \[ \log B_n\sim(\pi^2/12)n\tag{\(**\)} \] (Khintchine proved \((**)\) only with an undetermined constant instead of \(\pi^2/12\)) and the ``Ostrowski Algorithm'' described in [\textit{A. M. Rockett} and \textit{P. Szüsz}, Continued fractions Singapore: World Scientific, (1992; Zbl 0925.11038)].
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A05
zbMATH DE Number: 1800028
zbMATH Keywords: iteration
zbMATH Keywords: asymptotic relations
zbMATH Keywords: continued fraction
zbMATH Keywords: expansion
MaRDI profile type: Publication
full work available at URL: pjaa.77.134
OpenAlex ID: W2090726610
cites work: The Hausdorff dimension of recurrent sets in symbolic spaces
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DOI: 10.3792/PJAA.77.134
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description: scientific article; zbMATH DE number 1800028