# ANNALES DE L'INSTITUT FOURIER

On simultaneous rational approximation to a real number and its integral powers
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2165-2182.

For a positive integer $n$ and a real number $\xi$, let ${\lambda }_{n}\left(\xi \right)$ denote the supremum of the real numbers $\lambda$ such that there are arbitrarily large positive integers $q$ such that $||q\xi ||,||q{\xi }^{2}||,...,||q{\xi }^{n}||$ are all less than ${q}^{-\lambda }$. Here, $||·||$ 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 $\left({\lambda }_{1},...,{\lambda }_{n},...\right)$. We further address several open problems.

Pour un entier strictement positif $n$ et un nombre réel $\xi$, on note ${\lambda }_{n}\left(\xi \right)$ le supremum des nombres réels $\lambda$ pour lesquels il existe des entiers $q$ arbitrairement grands tels que $||q\xi ||,||q{\xi }^{2}||,...,||q{\xi }^{n}||$ sont tous inférieurs à ${q}^{-\lambda }$. Ici, $||·||$ 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 $\left({\lambda }_{1},...,{\lambda }_{n},...\right)$. Nous formulons également plusieurs problèmes ouverts.

DOI: 10.5802/aif.2580
Classification: 11J13
Keywords: Simultaneous approximation, exponent of approximation
Bugeaud, Yann 1

1 Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)
@article{AIF_2010__60_6_2165_0,
author = {Bugeaud, Yann},
title = {On simultaneous rational approximation to a real number and its integral powers},
journal = {Annales de l'Institut Fourier},
pages = {2165--2182},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {60},
number = {6},
year = {2010},
doi = {10.5802/aif.2580},
mrnumber = {2791654},
zbl = {1229.11100},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2580/}
}
TY  - JOUR
AU  - Bugeaud, Yann
TI  - On simultaneous rational approximation to a real number and its integral powers
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 2165
EP  - 2182
VL  - 60
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2580/
UR  - https://www.ams.org/mathscinet-getitem?mr=2791654
UR  - https://zbmath.org/?q=an%3A1229.11100
UR  - https://doi.org/10.5802/aif.2580
DO  - 10.5802/aif.2580
LA  - en
ID  - AIF_2010__60_6_2165_0
ER  - 
%0 Journal Article
%A Bugeaud, Yann
%T On simultaneous rational approximation to a real number and its integral powers
%J Annales de l'Institut Fourier
%D 2010
%P 2165-2182
%V 60
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2580
%R 10.5802/aif.2580
%G en
%F AIF_2010__60_6_2165_0
Bugeaud, Yann. On simultaneous rational approximation to a real number and its integral powers. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2165-2182. doi : 10.5802/aif.2580. https://aif.centre-mersenne.org/articles/10.5802/aif.2580/

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

[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., Volume 166 (2007), pp. 367-426 (with an appendix by R.C. Vaughan: ”Sums of two squares near perfect squares”) | DOI | MR | Zbl

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

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

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

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

[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)

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

[11] Bugeaud, Y.; Laurent, M. Exponents of Diophantine approximation, Diophantine Geometry Proceedings, Volume 4 (2007), pp. 101-121 | MR | Zbl

[12] Güting, R. Zur Berechnung der Mahlerschen Funktionen ${w}_{n}$, J. reine angew. Math., Volume 232 (1968), pp. 122-135 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

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

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

[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

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

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

Cited by Sources: