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 ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 6, p. 2165-2182
Pour un entier strictement positif n et un nombre réel ξ, on note λ n (ξ) le supremum des nombres réels λ pour lesquels il existe des entiers q arbitrairement grands tels que ||qξ||,||qξ 2 ||,...,||qξ n || sont tous inférieurs à q -λ . Ici, ||·|| désigne la distance à l’entier le plus proche. Nous étudions l’ensemble des valeurs prises par la function λ n et, plus généralement, nous nous intéressons au spectre de (λ 1 ,...,λ n ,...). Nous formulons également plusieurs problèmes ouverts.
For a positive integer n and a real number ξ, let λ n (ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ||qξ||,||qξ 2 ||,...,||qξ n || are all less than q -λ . Here, ||·|| denotes the distance to the nearest integer. We study the set of values taken by the function λ n and, more generally, we are concerned with the joint spectrum of (λ 1 ,...,λ n ,...). We further address several open problems.
DOI : https://doi.org/10.5802/aif.2580
Classification:  11J13
Mots clés: approximation simultanée, exposant d’approximation
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     pages = {2165-2182},
     doi = {10.5802/aif.2580},
     mrnumber = {2791654},
     zbl = {1229.11100},
     language = {en},
     url = {http://www.numdam.org/item/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, Tome 60 (2010) no. 6, pp. 2165-2182. doi : 10.5802/aif.2580. http://www.numdam.org/item/AIF_2010__60_6_2165_0/

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