Vahlen theorem for two-dimensional convergents
type: Item
instance of: scholarly article
title: Vahlen theorem for two-dimensional convergents
zbMATH Open document ID: 0981.11024
DOI: 10.1007/BF02674067
author: V. A. Bykovskiĭ
published in: Mathematical Notes
publication date: 2000-08-03T00:00:00Z
review text: Given \(\underline{\alpha}= (\alpha_1,\dots, \alpha_d)\in \mathbb{R}^d\), \(q\in \mathbb{N}\), let \(a_j\) be the nearest integers to the numbers \(qa_j\). A vector \((a_1,\dots, a_d;q)\) is called convergent of first type to \(\underline{\alpha}\) if the system of inequalities \(\|q'\alpha_j\|\leq \|q\alpha_j\|\) \((j= 1,\dots, d)\) is satisfied for no other positive integer \(q'< q\). Let \(\Delta(\underline{\alpha})\) denote the set of all ``denominators'' \(q\) that determine convergents uniquely. This set is infinite except when \(\underline{\alpha}\) is rational. Further, given \(\underline{\alpha}\in \mathbb{R}^d\), \({\mathbf q}= (q_1,\dots, q_d)\in \mathbb{Z}^d\), \({\mathbf q}\neq 0\), let \(a\) be the integer nearest to \({\mathbf {qa}}\). A vector \((a; q_1,\dots, q_d)\) is called convergent of second type to \(\alpha\) if for any other integer vector \({\mathbf q}'\neq 0\) with \(|q_j'|\leq |q_j|\) the following inequalities are satisfied: \[ \begin{aligned} \|{\mathbf {qa}}\|&< \|{\mathbf q}' {\mathbf a}\|\quad \text{for} \quad ((|q_1'|,\dots, |q_d'|)\neq (|q_1|,\dots, |q_d|), \qquad \text{and}\\ \|{\mathbf {qa}}\|&\leq \|{\mathbf q}' {\mathbf a}\|\quad \text{for} \quad ((|q_1'|,\dots, |q_d'|)= (|q_1|,\dots, |q_d|). \end{aligned} \] Let \(\Delta^* (\underline{\alpha})\) denote the set of all ``denominators'' \({\mathbf q}\) of convergents of the second type. The author proves the following results which both generalize Vahlen's theorem for ordinary continued fractions. (I) Given \(\underline{\alpha}= (\alpha_1, \alpha_2)\) and \(q\in \Delta(\underline{\alpha})\), let \(q'\in \Delta(\underline{\alpha})\) be the least denominator such that \(q< q'\), \(\|q'\alpha_1\|\leq \|q\alpha_1\|\), \(\|q'\alpha_2\|\leq \|q\alpha_2\|\). Then \[ \min \{q\|q\alpha_1\|\|q\alpha_2\|,\;q'\|q'\alpha_1\|\|q'\alpha_2\|\}< \tfrac{1}{2}. \] (II) Given \(\underline{\alpha}= (\alpha_1, \alpha_2)\) and \({\mathbf q}\in \Delta^*(\underline{\alpha})\), let \({\mathbf q}'\in \Delta…
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 1429646
zbMATH Keywords: simultaneous Diophantine approximation
zbMATH Keywords: convergents
zbMATH Keywords: Vahlen's theorem
zbMATH Keywords: continued fractions
reviewed by: Gerhard Ramharter
MaRDI profile type: Publication
cites work: Q5603911
cites work: Q3241280
full work available at URL: https://doi.org/10.1007/bf02674067
OpenAlex ID: W2011980290
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description: scientific article; zbMATH DE number 1429646