Diophantine aspects of the Calkin-Wilf iteration
type: Item
instance of: scholarly article
title: Diophantine aspects of the Calkin-Wilf iteration
zbMATH Open document ID: 1312.11006
DOI: 10.4171/EM/170
author: J. W. Sander
author: Jörn Steuding
author: R. Steuding
published in: Elemente der Mathematik
publication date: 2011-07-12T00:00:00Z
review text: The Calkin-Wilf tree is defined as a binary tree with a root \(1/1\) and the left and right child of a node \(a/b\) are \(a/(a+b)\) and \((a+b)/b\), respectively. In this paper, the authors studied new diophantine aspects of this tree as follows. Let \(U(z)=z+1\) and \(T(z)=z+1\) be two Möbius transformations. For given \(\alpha\) a complex number, the \(\alpha\)-Calkin-Wilf tree is defined as a binary tree with a root \(\alpha\) and the left and right child of a node \(z\) are \(U(z)\) and \(T(z)\), respectively. The authors described the relation of the \(\alpha\)-Calkin-Wilf iteration to continued fractions. Then they showed that the \(\alpha\)-Calkin-Wilf tree contains any positive number \(\beta\) of the form \(\beta=\frac{a\alpha+b}{c\alpha+d}\) with \(a,b,c,d\geq1\) and \(ad-bc=1\). all these numbers \(\beta\) have the same tail in their continued fraction expansion as \(\alpha\).
reviewed by: Toufik Mansour
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 5919429
zbMATH Keywords: Calkin-Wilf tree
zbMATH Keywords: continued fraction
zbMATH Keywords: Möbius transformation
describes a project that uses: OEIS
MaRDI profile type: Publication
Wikidata QID: Q60692787
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full work available at URL: https://doi.org/10.4171/em/170
OpenAlex ID: W2090591319
description: scientific article; zbMATH DE number 5919429