Linear fractional transformations of continued fractions with bounded partial quotients
type: Item
instance of: scholarly article
title: Linear fractional transformations of continued fractions with bounded partial quotients
zbMATH Open document ID: 0901.11024
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-11-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93691
full work available at URL: jtnb9-2.html
review text: Let \(\theta\) be a real number with continued fraction expansion \(\theta=[a_0,a_1,a_2,\dots]\), and let \(M=\left[\begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right]\) be an integer matrix with \(\det(M)\neq 0\). Set \(K(\theta)=\sup_{i\geq 1}a_i\) and \(K_\infty\limsup_{i\geq 1}a_i\). Then \[ {1\over| \det(M)| }K_\infty(\theta)-2\leq K_\infty \Biggl({a\theta+b\over c\theta+d}\Biggr) \leq| \det(M)| (K_\infty(\theta)+2) \] and \[ K\Biggl( {a\theta+b\over c\theta+d} \Biggr)\leq| \det(M)| (K(\theta)+2)+| c(c\theta+d)| . \] The proofs use the type \(L(\theta)=\sup_{q\geq 1}(q\| q\theta\|)^{-1}\) and the Lagrange constant \[ L_\infty(\theta)= \limsup_{q\geq 1}(q\| q\theta\|)^{-1} \] for an irrational \(\theta\). The former follows from the facts \(L_{\infty}({a\theta+b\over c\theta+d})=L_{\infty}(\theta)\), and \(\sup(L_{\infty}({a\theta+b\over c\theta+d}))=m L_{\infty}(\theta)\) and \(\inf(L_{\infty}({a\theta+b\over c\theta+d}))\geq{1\over m}L_{\infty}(\theta)\), where \(M\) takes all the matrices such that \(| \det(M)| =m\). The latter follows from the result \[ L\Biggl( {a\theta+b\over c\theta+d} \Biggr)\leq| \det(M)| L(\theta)+| c(c\theta+d)| . \] The authors also show as one of the remarks that \[ {1\over| \det(M)| }L_\infty(\theta)\leq L_\infty \Biggl({a\theta+b\over c\theta+d}\Biggr) \leq| \det(M)| L_\infty(\theta). \]
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11J06
zbMATH DE Number: 1164087
zbMATH Keywords: Lagrange spectrum
zbMATH Keywords: diophantine approximation
zbMATH Keywords: linear fractional transformation
zbMATH Keywords: continued fractions
zbMATH Keywords: bounded partial quotients
author: Jeffrey C. Lagarias
author: Jeffrey Shallit
reviewed by: Takao Komatsu
MaRDI profile type: Publication
OpenAlex ID: W2093972771
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DOI: 10.5802/JTNB.202
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description: scientific article; zbMATH DE number 1164087