On the approximation by continued fractions. II
type: Item
instance of: scholarly article
title: On the approximation by continued fractions. II
zbMATH Open document ID: 0702.11039
DOI: 10.1016/0019-3577(90)90033-J
published in: Indagationes Mathematicae. New Series
publication date: 1990-01-01T00:00:00Z
review text: With the help of a powerful method developed by \textit{H. Jager} and the author [Indag. Math. 51, 289-307 (1989; Zbl 0695.10029)] which is based on the dynamical system underlying the continued fraction algorithm, the author gives short proofs of already known results on asymmetric diophantine approximation and he also gives answers to some open questions on this topic as for example the following solution of a problem settled by Tong. Theorem: Let \(\tau\geq 1\) and let x be a real irrational number with continued fraction expansion \(x=[a_ 0,a_ 1,a_ 2,...]\). Then from the three best approximation convergents \(p_ i/q_ i\), \(i=n,n+1,n+2\), \(n\geq 1\), at least one satisfies \[ -\frac{\tau}{\sqrt{a^ 2_{n+2}+4\tau}}\cdot \frac{1}{q^ 2_ i}<x-\frac{p_ i}{q_ i}<\frac{1}{\sqrt{a^ 2_{n+2}+4\tau}}\cdot \frac{1}{q^ 2_ i}. \] In case \(0<\tau <1\) there exists an irrational number x and a positive integer n such that \(p_ i/q_ i\) does not satisfy the inequality for \(i=2n,2n+1,2n+2\).
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 4150290
zbMATH Keywords: approximation constants
zbMATH Keywords: continued fraction
zbMATH Keywords: asymmetric diophantine approximation
zbMATH Keywords: best approximation convergents
author: Cor Kraaikamp
reviewed by: Gerhard Larcher
MaRDI profile type: Publication
cites work: Q3715211
cites work: Q3470544
cites work: On asymmetric approximations
cites work: Unsymmetrical approximation of irrational numbers
cites work: Lattice points in infinite domains and asymmetric diophantine approximations
cites work: On a theorem of Segre
cites work: Segre's theorem on asymmetric diophantine approximation
cites work: A theorem on approximation of irrational numbers by simple continued fractions
full work available at URL: 0019-3577(90)90033-j
OpenAlex ID: W4205736544
description: scientific article; zbMATH DE number 4150290