For a positive integer and a real number , let denote the supremum of the real numbers such that there are arbitrarily large positive integers such that are all less than . Here, denotes the distance to the nearest integer. We study the set of values taken by the function and, more generally, we are concerned with the joint spectrum of . We further address several open problems.
Pour un entier strictement positif et un nombre réel , on note le supremum des nombres réels pour lesquels il existe des entiers arbitrairement grands tels que sont tous inférieurs à . Ici, désigne la distance à l’entier le plus proche. Nous étudions l’ensemble des valeurs prises par la function et, plus généralement, nous nous intéressons au spectre de . Nous formulons également plusieurs problèmes ouverts.
Keywords: Simultaneous approximation, exponent of approximation
Mot 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}, pages = {2165--2182}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {6}, year = {2010}, doi = {10.5802/aif.2580}, zbl = {1229.11100}, mrnumber = {2791654}, language = {en}, url = {http://archive.numdam.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 - http://archive.numdam.org/articles/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 http://archive.numdam.org/articles/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. http://archive.numdam.org/articles/10.5802/aif.2580/
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