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 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.

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.

DOI: 10.5802/aif.2580
Classification: 11J13
Keywords: Simultaneous approximation, exponent of approximation
Mot clés : approximation simultanée, exposant d’approximation
Bugeaud, Yann 1

1 Université de Strasbourg Mathématiques 7, rue René Descartes 67084 Strasbourg (FRANCE)
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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/

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