Linear forms at a basis of an algebraic number field
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 \(n=2\), the author improves to: Let \((1,\alpha_1,\alpha_2)\) be a basis of a real cubic field \(E\) over \(\mathbb Q\). Then there exists a positive real constant \(\kappa\) for which there are arbitrarily large integers \(M\), and non-zero integers \(x_0,x_1x_2\) with \(\max\{|x_0|, |x_1|, |x_2|\}=M\), \(|x_2|\ll M\log\;M^{-\kappa}\) and \(|x_0+x_1\alpha_1+x_2\alpha_2|\ll M^{-2}\). In relation with the dual form of Schmidt's theorem [\textit{W. M. Schmidt}, ``Approximation to algebraic numbers'', Enseign. Math., II. Sér. 17, 187--253 (1971; Zbl 0226.10033)], he proves: Assume that \((1,\alpha_1,\dots,\alpha_n)\) is a basis of an algebraic number field \(E\), with \(n\geq 2\). If \(\mathcal S\) is an infinite set of \((n+1)\)-tuples \((x_0,\dots,x_n)\in \mathbb Z^{n+1}\) satisfying \(|x_0+x_1\alpha_1+\dots+x_n\alpha_n|\ll M^{-n}\) and \(\max_{\;0\leq i\leq n}|x_i|=M\) with \(M\geq 2\), then there exists a positive real constant \(\lambda\) such that we have, for each \((x_0,\dots,x_n)\in\mathcal S\), \(\max_{\;2\leq i\leq n}|x_i|\gg M\log M^{-\lambda}\). Moreover if \(\mathbb Q(\alpha_1)\not=E\), then \(\max_{\;2\leq i\leq n} |x_i|\gg M\).
scientific article; zbMATH DE number 5966261
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