Linear forms at a basis of an algebraic number field
type: Item
instance of: scholarly article
title: Linear forms at a basis of an algebraic number field
zbMATH Open document ID: 1254.11069
author: Bernard de Mathan
published in: Journal of Number Theory
publication date: 2011-11-02T00:00:00Z
review text: For a real number \(x\), let us denote \(\|x\|\) the distance between \(x\) and the nearest integer. \textit{Littlewood conjecture} asserts that: \textit{for any real numbers \(\alpha\) and \(\beta\), one has \[ \inf_{q>0} q\|q\alpha\|\|q\beta\|=0, \] where \(q\) runs through the positive integers.} The \textit{dual form} of Littlewood conjecture is : \textit{for any real numbers \((\alpha_1,\dots,\alpha_n)\) one has \[ \inf_{x_1,\dots,x_n}\max\{|x_1|,1\}\dots\max\{|x_n|,1\}\|x_1\alpha_1+\dots+x_n\alpha_n\|=0. \] } It was proved by \textit{J. W. S. Cassels} and \textit{H. P. F. Swinnerton-Dyer} [ ``On the product of three homogeneous linear forms and indefinite ternary quadratic forms'', Philos. Trans. R. Soc. Lond., Ser. A 248, 73--96 (1955; Zbl 0065.27905)] that the Littlewood conjecture holds for any pair of numbers in a cubic field. Later, this result was generalized by \textit{L. G. Peck} [``Simultaneous rational approximations to algebraic numbers'', Bull. Am. Math. Soc. 67, 197--201 (1961; Zbl 0098.26302)] to a basis \((1,\alpha_1,\dots,\alpha_n)\) of a real algebraic number field of degree at least \(3\). This result provides some case of the dual form of the Littlewood conjecture. By a generalization of Peck's method, the author finds another case and, using Baker's estimates for linear forms in logarithms of algebraic numbers [\textit{A. Baker}, ``A sharpening of the bounds for linear forms in logarithms'', Acta Arith. 21, 117--129 (1972; Zbl 0244.10031)], he discusses whether this result is best possible. More precisely, by an inhomogeneous version of Peck method, he proves: Let \((1,\alpha_1,\dots,\alpha_n)\) be a basis of a real algebraic number field \(E\) of degree \(n+1\geq 3\) over \(\mathbb Q\). Then there exist infinitely many positive integers \(M\) for which there are integers \(x_0,\dots,x_n\), not all zero, such that \(\max_{\;0\leq i\leq n}|x_i|=M\), \(|x_n|=o(M)\) and \(|x_0+x_1\alpha_1+\dots+x_n\alpha_n|\ll M^{-n}\). In the case \(n=2\), th…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5966261
zbMATH Keywords: Simultaneous diophantine approximation
zbMATH Keywords: Littlewood conjecture
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.06.009
OpenAlex ID: W2093632254
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DOI: 10.1016/J.JNT.2011.06.009
description: scientific article; zbMATH DE number 5966261