no. 5
Meilleures approximations diophantiennes simultanées et théorème de Lévy
Chevallier, Nicolas1
1 Université de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse (France)
Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1635-1657.

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.

DOI : 10.5802/aif.2134
Classification : 11J13, 11J70, 22F30
Mot clés : approximations diophantiennes, théorème de Lévy, réseaux
Keywords: Diophantine approximations, Lévy's theorem, lattices

Chevallier, Nicolas 1

1 Université de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse (France) 
@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},
     zbl = {1080.11052},
     mrnumber = {2172275},
     language = {fr},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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. https://aif.centre-mersenne.org/articles/10.5802/aif.2134/

[Be, Ma] M.B. Bekka; M. Mayer Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, Lecture Note Series, 269, London Mathematical Society, 2000 | Zbl

[Br,Gu] A. Broise-Alamichel; Y. Guivarc'h Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée, Annales Institut Fourier, Volume 51 (2001) no. 3, pp. 565-686 | DOI | Numdam | MR | Zbl

[Br1] A.J. Brentjes A two-dimensional continued fraction algorithm for best approximations with an application in cubic fields, J. Reine Angew. Math., Volume 326 (1981), pp. 18-44 | MR | Zbl

[Br2] A.J. Brentjes Multi-dimensional continued fraction algorithms, Mathematical Center Tracts, 145, Math. Centrum, Amsterdam, 1981 | Zbl

[Ca] J.W.S. Cassels An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, 45, Cambridge Univ. Press, 1965 | Zbl

[Ch1] N. Chevallier Distances dans la suite des multiples d'un point du tore à deux dimensions, Acta Arith., Volume 74 (1996), pp. 47-59 | MR | Zbl

[Ch2] N. Chevallier Meilleures approximations d'un élément du tore 𝕋 2 et géométrie de la suite des multiples de cet élément, Acta Arith., Volume 78 (1996), pp. 19-35 | MR | Zbl

[Ch3] N. Chevallier Géométrie des suites de Kronecker, Manuscripta Math., Volume 94 (1997), pp. 231-241 | DOI | MR | Zbl

[Ch4] N. Chevallier Meilleures approximations diophantiennes d'un élément du tore 𝕋 d , Acta Arith., Volume 97 (2001), pp. 219-240 | DOI | MR | Zbl

[Ch5] N. Chevallier Meilleures approximations diophantiennes simultanées, Cahiers du séminaire de probabilités, Rennes (2002)

[Da] S.G. Dani Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89 | MR | Zbl

[Gr, Lag] D.J. Grabiner; J.C. Lagarias Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions, Monatsh, Volume 133 (2001), pp. 295-339 | MR | Zbl

[Lag1] J.C. Lagarias; P. Ribenboim Some new results in simultaneous diophantine approximation, Proc. of the Queen's Number Theory Conference 1979 (Queen's Paper in Pure and Applied Math.), Volume 54 (1980), pp. 453-474 | Zbl

[Lag2] J.C. Lagarias Best simultaneous diophantine approximations I, Growth Rates of Best Approximations denominators, Trans. Amer. Math. Soc., Volume 272 (1982) no. 2, pp. 545-554 | MR | Zbl

[Lag3] J.C. Lagarias Best simultaneous diophantine approximations II, behavior of consecutive best approximations, Pacific Journal of Mathematics, Volume 102 (1982) no. 1, pp. 61-88 | MR | Zbl

[Lag4] J.C. Lagarias Best diophantine approximations to a set of linear forms, J. Austral. Math. Soc. Ser. A, Volume 34 (1983), pp. 114-122 | DOI | MR | Zbl

[Lag5] J.C. Lagarias Geodesic multidimensional continued fractions, Proc. London Math. Soc., Volume 69 (1994) no. 3, pp. 464-488 | DOI | MR | Zbl

[Ma] G.A. Margulis; Wüstholz Diophantine Approximations, Lattices and flows on Homogeneous Spaces, A panorama of number theory or the view from Baker garden (2002), pp. 280-310 | Zbl

[Ro] C.A. Rogers The signature of the errors of some simultaneous Diophantine approximations, Proc. London Math. Soc., Volume 52 (1951), pp. 186-190 | MR | Zbl

[Schm] W.M. Schmidt A metrical theorem in diophantine approximation, Canadian J. Math., Volume 12 (1960), pp. 619-631 | DOI | MR | Zbl

[Schw] F. Schweiger Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, 2000 | MR | Zbl

[Sp] V.G. Sprind\v {z}uk Metric Theory of Diophantine Approximations, W.H. Winston Sons, Washington, C. D., 1979 | Zbl

[Wa] F.W. Warner Foundations of Differentiable Manifolds and Lie Groups, G.T.M., 94, Springer-Verlag, 1983 | MR | Zbl

Cité par Sources :