Approximation by algebraic numbers
type: Item
instance of: scholarly article
title: Approximation by algebraic numbers
zbMATH Open document ID: 0967.11025
author: Yann Bugeaud
published in: Journal of Number Theory
publication date: 2001-02-06T00:00:00Z
review text: Let \(m\geq 1\) and \(d\geq 2\) be integers, \(\varepsilon\), \(\lambda\) real numbers with \(0<\varepsilon<\lambda<d\) and \(\theta\) an algebraic number of degree \(>d\). There exists a real number \(H_{0}\) with the following property: For each \(H\geq H_{0}\) there is an algebraic number \(\alpha\) of degree \(d\) and usual height \(H(\alpha)\) with \[ H^{1-(\varepsilon/\lambda)}\leq H(\alpha)\leq H \quad\text{ and }\quad |\theta-\alpha|\leq H(\alpha)^{-d-1+\varepsilon}. \] This refines earlier results of \textit{E. Wirsing} [J. Reine Angew. Math. 206, 67-77 (1961; Zbl 0097.03503)] and \textit{W. M. Schmidt} [Enseign. Math., II. Sér. 17, 187--253 (1971; Zbl 0226.10033)] in two ways: the lower bound for \(H(\alpha)\) is new, as well as the information that \(\alpha \) has degree \(d\) (and not merely \(\leq d\)). A similar result is shown to be true for almost all real numbers \(\theta\) (for Lebesgue measure). Also for almost all real (resp. complex) numbers \(\theta\), the author refines the result of \textit{M.~Laurent} and \textit{D.~Roy} [Ann. Inst. Fourier 49, No. 1, 27--55 (1999; Zbl 0923.11105)] on the existence of algebraic approximation by algebraic numbers: He provides lower bounds for the degree as well as for the height of the approximants. He shows that such statements do not hold for \(U\)-numbers. Further related results are established, including a refinement of an estimate due to \textit{E.~Bombieri} and \textit{J.~Mueller} [Mich. Math. J. 33, 83--93 (1986; Zbl 0593.10031)] providing an effective improvement of Liouville's inequality.
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11J04
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 1533153
zbMATH Keywords: lower bound
zbMATH Keywords: algebraic approximation by algebraic numbers
zbMATH Keywords: degree
zbMATH Keywords: height
zbMATH Keywords: effective improvement of Liouville's inequality
MaRDI profile type: Publication
full work available at URL: jnth.2000.2517
OpenAlex ID: W2066390705
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DOI: 10.1006/JNTH.2000.2517
description: scientific article; zbMATH DE number 1533153