On simultaneous rational approximation to a real number and its integral powers

[Sur l’approximation rationnelle simultanée d’un nombre réel et de ses puissances entières]

Bugeaud, Yann

Annales de l'Institut Fourier, Tome 60 (2010), p. 2165-2182 / Harvested from Numdam

Résumé

Pour un entier strictement positif $n$ et un nombre réel $\xi $, on note $\lambda _n (\xi )$ le supremum des nombres réels $\lambda $ pour lesquels il existe des entiers $q$ arbitrairement grands tels que $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ sont tous inférieurs à $q^{-\lambda }$. Ici, $|| \cdot ||$ désigne la distance à l’entier le plus proche. Nous étudions l’ensemble des valeurs prises par la function $\lambda _n$ et, plus généralement, nous nous intéressons au spectre de $(\lambda _1, \ldots , \lambda _n , \ldots )$. Nous formulons également plusieurs problèmes ouverts.

Abstract

For a positive integer $n$ and a real number $\xi $, let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda $ such that there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi ^2 ||, \ldots , ||q \xi ^n||$ are all less than $q^{-\lambda }$. Here, $|| \cdot ||$ denotes the distance to the nearest integer. We study the set of values taken by the function $\lambda _n$ and, more generally, we are concerned with the joint spectrum of $(\lambda _1, \ldots , \lambda _n , \ldots )$. We further address several open problems.

Détail

Publié le : 2010-01-01

MR 2791654 | Zbl 1229.11100

DOI : https://doi.org/10.5802/aif.2580
Classification:  11J13
Mots clés: approximation simultanée, exposant d’approximation

Bibliographie

[1] Adamczewski, B.; Bugeaud, Y. Palindromic continued fractions, Ann. Inst. Fourier (Grenoble), Tome 57 (2007), pp. 1557-1574 | Article | Numdam | MR 2364142 | Zbl 1126.11036

[2] Beresnevich, V. Rational points near manifolds and metric Diophantine approximation (preprint)

[3] Beresnevich, V.; Dickinson, D.; Velani, S. L. Diophantine approximation on planer curves and the distribution of rational points, Ann. of Math., Tome 166 (2007), pp. 367-426 (with an appendix by R.C. Vaughan: ”Sums of two squares near perfect squares”) | Article | MR 2373145 | Zbl 1137.11048

[4] Bernik, V. I. Application of the Hausdorff dimension in the theory of Diophantine approximations, Acta Arith., Tome 42 (1983), pp. 219-253 ((in Russian), english transl. in Amer. Math. Soc. Transl. 140 (1988), p. 15–44) | MR 729734 | Zbl 0482.10049

[5] Budarina, N.; Dickinson, D.; Levesley, J. Simultaneous Diophantine approximation on polynomial curves, Mathematika, Tome 56 (2010), pp. 77-85 | Article | MR 2604984 | Zbl pre05673374

[6] Bugeaud, Y. Approximation by algebraic numbers, Cambridge University Press, Cambridge Tracts in Mathematics (2004) | MR 2136100 | Zbl 1055.11002

[7] Bugeaud, Y. Diophantine approximation and Cantor sets, Math. Ann., Tome 341 (2008), pp. 677-684 | Article | MR 2399165 | Zbl 1163.11056

[8] Bugeaud, Y. Multiplicative Diophantine approximation, Dynamical systems and Diophantine Approximation (to appear) (proceedings of the conference held at the Institut Henri Poincaré, S.M.F.)

[9] Bugeaud, Y.; Laurent, M. On transfer inequalities in Diophantine approximation, II (Math. Z., to appear) | Zbl pre05708689

[10] Bugeaud, Y.; Laurent, M. Exponents of Diophantine Approximation and Sturmian Continued Fractions, Ann. Inst. Fourier (Grenoble), Tome 55 (2005), pp. 773-804 | Article | Numdam | MR 2149403 | Zbl 1155.11333

[11] Bugeaud, Y.; Laurent, M. Exponents of Diophantine approximation, Diophantine Geometry Proceedings, Scuola Normale Superiore Pisa, Ser. CRM, Tome 4 (2007), pp. 101-121 | MR 2349650 | Zbl 1229.11098 | Zbl pre05263280

[12] Güting, R. Zur Berechnung der Mahlerschen Funktionen $w_n$, J. reine angew. Math., Tome 232 (1968), pp. 122-135 | Article | MR 233776 | Zbl 0174.08503

[13] Jarník, V. Über die simultanen Diophantische Approximationen, Math. Z., Tome 33 (1931), pp. 505-543 | Article | MR 1545226

[14] Jarník, V. Über einen Satz von A. Khintchine II, Acta Arith., Tome 2 (1936), pp. 1-22 | Zbl 0015.29405

[15] Kleinbock, D.; Lindenstrauss, E.; Weiss, B. On fractal measures and Diophantine approximation, Selecta Math., Tome 10 (2004), pp. 479-523 | MR 2134453 | Zbl 1130.11039

[16] Lang, S. Algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 211 (2002) | MR 1878556 | Zbl 0984.00001

[17] Laurent, M. On transfer inequalities in Diophantine Approximation, Analytic Number Theory in Honour of Klaus Roth, Cambridge University Press (2009), pp. 306-314 | MR 2508652 | Zbl 1163.11053

[18] Mahler, K. Zur Approximation der Exponentialfunktionen und des Logarithmus. I, II, J. reine angew. Math., Tome 166 (1932), pp. 118-150 | Article | Zbl 0003.38805

[19] Schmidt, W. M. On heights of algebraic subspaces and diophantine approximations, Annals of Math., Tome 85 (1967), pp. 430-472 | Article | MR 213301 | Zbl 0152.03602

[20] Sprindžuk, V. G. Mahler’s problem in metric number theory, Nauka i Tehnika, Minsk (1967) ((in Russian), english translation by B. Volkmann, Translations of Mathematical Monographs, Vol. 25, American Mathematical Society, Providence, R.I., 1969) | Zbl 0168.29504

[21] Vaughan, R. C.; Velani, S. Diophantine approximation on planar curves: the convergence theory, Invent. Math., Tome 166 (2006), pp. 103-124 | Article | MR 2242634 | Zbl 1185.11047

[22] Wirsing, E. Approximation mit algebraischen Zahlen beschränkten Grades, J. reine angew. Math., Tome 206 (1961), pp. 67-77 | Article | MR 142510 | Zbl 0097.03503