Segre's theorem on asymmetric diophantine approximation
type: Item
instance of: scholarly article
title: Segre's theorem on asymmetric diophantine approximation
zbMATH Open document ID: 0645.10009
DOI: 10.1016/0022-314X(88)90122-9
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: The following theorem is proved: Let \(\xi\) be a real irrational number, \(p_i/q_i\) \((i=1,2,\ldots)\) the convergents of its regular continued fraction expansion. Then we have for any even \(n\ge 2\) \[ -(a^2_{n+1}+4\tau)^{-} < (-1)^{n+1}q^2_1(\xi - p_i/q_i) < \tau(a^2_{n+1}+4\tau)^{-} \] at least once for \(i=n-1,n,n+1\); here \(a_{i+1}\) is the \(n+1\)st partial quotient of the continued fraction expansion of \(\xi\) and \(\tau\) an arbitrary positive number. As corollaries of the above theorem the proof of a conjecture of LeVeque and a proof of the well-known fact are given that any irrational number not equivalent to \((5^{1/2}-1)/2\) admits a rational approximation \(| \xi - p/q| < (8^{1/2}q^ 2)^{-1}\).
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11J04
Mathematics Subject Classification ID: 11A55
zbMATH DE Number: 4051713
zbMATH Keywords: asymmetric Diophantine approximation
zbMATH Keywords: real irrational number
zbMATH Keywords: convergents
zbMATH Keywords: regular continued fraction expansion
author: Jing Cheng Tong
MaRDI profile type: Publication
cites work: Q5507792
cites work: On asymmetric approximations
cites work: Q5723292
cites work: Lattice points in infinite domains and asymmetric diophantine approximations
cites work: On a theorem of Segre
cites work: The conjugate property of the Borel theorem on diophantine approximation
cites work: A theorem on approximation of irrational numbers by simple continued fractions
description: scientific article; zbMATH DE number 4051713