D'après le théorème de Lévy, les dénominateurs du développement en fraction continue d'un réel croissent presque sûrement à une vitesse au plus exponentielle. Nous étendons cette estimation aux meilleures approximations diophantiennes simultanées de formes linéaires.
According to Lévy's theorem, the denominators of the continued fraction expansion of a real number almost surely grow at most at the rate of a geometric series. We extend this estimate to best simultaneous Diophantine approximations to a set of linear forms.
Mot clés : approximations diophantiennes, théorème de Lévy, réseaux
Keywords: Diophantine approximations, Lévy's theorem, lattices
@article{AIF_2005__55_5_1635_0,
author = {Chevallier, Nicolas},
title = {Meilleures approximations diophantiennes simultan\'ees et th\'eor\`eme de {L\'evy}},
journal = {Annales de l'Institut Fourier},
pages = {1635--1657},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {55},
number = {5},
year = {2005},
doi = {10.5802/aif.2134},
mrnumber = {2172275},
zbl = {1080.11052},
language = {fr},
url = {http://www.numdam.org/articles/10.5802/aif.2134/}
}
TY - JOUR AU - Chevallier, Nicolas TI - Meilleures approximations diophantiennes simultanées et théorème de Lévy JO - Annales de l'Institut Fourier PY - 2005 SP - 1635 EP - 1657 VL - 55 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2134/ DO - 10.5802/aif.2134 LA - fr ID - AIF_2005__55_5_1635_0 ER -
%0 Journal Article %A Chevallier, Nicolas %T Meilleures approximations diophantiennes simultanées et théorème de Lévy %J Annales de l'Institut Fourier %D 2005 %P 1635-1657 %V 55 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2134/ %R 10.5802/aif.2134 %G fr %F AIF_2005__55_5_1635_0
Chevallier, Nicolas. Meilleures approximations diophantiennes simultanées et théorème de Lévy. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1635-1657. doi : 10.5802/aif.2134. http://www.numdam.org/articles/10.5802/aif.2134/
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