×

Simultaneous approximation by algebraic numbers. (Approximation simultanée par des nombres algébriques.) (French) Zbl 1071.11044

Let \(\xi\) be a transcendental complex number. For each integer \(n\geq 1\), denote by \(w_{n}^{\star}(\xi)\) the supremum of the set of real numbers \(w^{\star}\) such that there exist infinitely many algebraic numbers \(\alpha\), of degree \(\leq n\), satisfying \[ 0<| \xi-\alpha| \leq H(\alpha)^{-w^{\star}-1}, \] where \(H(\alpha)\) is the usual height of \(\alpha\). These numbers were introduced by J. F. Koksma in his classification of transcendental numbers. Next denote by \(\lambda_{n}(\xi)\) the minimum of \(d/(w_{d}^{\star}(\xi)+d)\), where \(d\) ranges over the set of integers \(1\leq d\leq n\) for which \(w_{d}^{\star}(\xi)+d+1\geq 3n/14\). The main result of this paper is that, for \(n\geq 8\), for each sufficiently large integer \(H\), there exists an algebraic number \(\alpha\) of degree \(\leq n\) and height \(\leq H\) such that \[ | \xi-\alpha| \leq\exp\left\{- \frac{n}{70} \lambda_{n}(\xi)\log H \right\}. \] The author deduces several consequences related to simultaneous approximation of transcendental numbers by algebraic numbers.
[Note of the reviewer: At the end of section 2 p. 669, replace twice Corollaire 1 by Corollaire 3].

MSC:

11J68 Approximation to algebraic numbers
11J13 Simultaneous homogeneous approximation, linear forms
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Bugeaud, Y., Approximation par des nombres algébriques. J. Number Theory84 (2000), 15-33. · Zbl 0967.11025
[2] Bugeaud, Y., Teulié, O., Approximation d’un nombre réel par des nombres algébriques de degré donné. Acta Arith.93 (2000), 77-86. · Zbl 0948.11029
[3] Diaz, G., Une nouvelle propriété d’approximation diophantienne. C. R. Acad. Sci. Paris324 (1997), 969-972. · Zbl 0899.11033
[4] Güting, R., Zur Berechnung der Mahlerschen Funktionen wn. J. reine angew. Math.232 (1968), 122-135. · Zbl 0174.08503
[5] Laurent, M., Roy, D., Criteria of algebraic independence with multiplicities and interpolation determinants. Trans. Amer. Math. Soc.351 (1999), 1845-1870. · Zbl 0923.11106
[6] Laurent, M., Roy, D., Sur l’approximation algébrique en degré de transcendance un. Annales Instit. Fourier49 (1999), 27-55. · Zbl 0923.11105
[7] Roy, D., Waldschmidt, M., Approximation diophantienne et indépendance algébrique de logarithmes. Ann. Sci. École Norm. Sup.30 (1997), 753-796. · Zbl 0895.11030
[8] Roy, D., Waldschmidt, M., Diophantine approximation by conjugate algebraic integers. A paraître. · Zbl 1055.11043
[9] Tishchenko, K.I., On simultaneous approximation of two real numbers by roots of the same polynomial. Preprint. · Zbl 1177.11056
[10] Schneider, T., Introduction aux nombres transcendants. Gauthier-Villars, Paris, 1959. · Zbl 0098.26304
[11] Wirsing, E., Approximation mit algebraischen Zahlen beschränkten Grades. J. reine angew. Math.206 (1961), 67-77. · Zbl 0097.03503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.