Best simultaneous Diophantine approximations and Lévy's theorem

Nicolas Chevallier[1]

  • [1] Université de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 5, page 1635-1657
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Chevallier, Nicolas. "Meilleures approximations diophantiennes simultanées et théorème de Lévy." Annales de l’institut Fourier 55.5 (2005): 1635-1657. <http://eudml.org/doc/116227>.

@article{Chevallier2005,
abstract = {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.},
affiliation = {Université de Haute Alsace, 4 rue des frères Lumière, 68093 Mulhouse (France)},
author = {Chevallier, Nicolas},
journal = {Annales de l’institut Fourier},
keywords = {Diophantine approximations; Lévy's theorem; lattices},
language = {fre},
number = {5},
pages = {1635-1657},
publisher = {Association des Annales de l'Institut Fourier},
title = {Meilleures approximations diophantiennes simultanées et théorème de Lévy},
url = {http://eudml.org/doc/116227},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Chevallier, Nicolas
TI - Meilleures approximations diophantiennes simultanées et théorème de Lévy
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 5
SP - 1635
EP - 1657
AB - 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.
LA - fre
KW - Diophantine approximations; Lévy's theorem; lattices
UR - http://eudml.org/doc/116227
ER -

References

top
  1. M.B. Bekka, M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, 269 (2000), London Mathematical Society Zbl0961.37001
  2. A.J. Brentjes, A two-dimensional continued fraction algorithm for best approximations with an application in cubic fields, J. Reine Angew. Math. 326 (1981), 18-44 Zbl0452.10034MR622343
  3. A.J. Brentjes, Multi-dimensional continued fraction algorithms, 145 (1981), Math. Centrum, Amsterdam Zbl0474.10024
  4. A. Broise-Alamichel, Y. Guivarc'h, Exposants caractéristiques de l'algorithme de Jacobi-Perron et de la transformation associée, Annales Institut Fourier 51 (2001), 565-686 Zbl1012.11060MR1838461
  5. J.W.S. Cassels, An Introduction to Diophantine Approximation, 45 (1965), Cambridge Univ. Press Zbl0077.04801
  6. N. Chevallier, Distances dans la suite des multiples d'un point du tore à deux dimensions, Acta Arith. 74 (1996), 47-59 Zbl0838.11052MR1367577
  7. N. Chevallier, Meilleures approximations d’un élément du tore 𝕋 2 et géométrie de la suite des multiples de cet élément, Acta Arith. 78 (1996), 19-35 Zbl0863.11043MR1424999
  8. N. Chevallier, Géométrie des suites de Kronecker, Manuscripta Math. 94 (1997), 231-241 Zbl0893.11028MR1473898
  9. N. Chevallier, Meilleures approximations diophantiennes d’un élément du tore 𝕋 d , Acta Arith. 97 (2001), 219-240 Zbl0984.11033MR1826002
  10. N. Chevallier, Meilleures approximations diophantiennes simultanées, Cahiers du séminaire de probabilités, Rennes (2002) 
  11. D.J. Grabiner, J.C. Lagarias, Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions, Monatsh 133 (2001), 295-339 Zbl1040.37008MR1915877
  12. J.C. Lagarias, Some new results in simultaneous diophantine approximation, Proc. of the Queen's Number Theory Conference 1979 54 (1980), 453-474 Zbl0453.10035
  13. S.G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55-89 Zbl0578.22012MR794799
  14. J.C. Lagarias, Best simultaneous diophantine approximations I, Growth Rates of Best Approximations denominators, Trans. Amer. Math. Soc. 272 (1982), 545-554 Zbl0495.10021MR662052
  15. J.C. Lagarias, Best simultaneous diophantine approximations II, behavior of consecutive best approximations, Pacific Journal of Mathematics 102 (1982), 61-88 Zbl0497.10025MR682045
  16. J.C. Lagarias, Best diophantine approximations to a set of linear forms, J. Austral. Math. Soc. Ser. A 34 (1983), 114-122 Zbl0498.10019MR683183
  17. J.C. Lagarias, Geodesic multidimensional continued fractions, Proc. London Math. Soc. 69 (1994), 464-488 Zbl0813.11040MR1289860
  18. G.A. Margulis, Diophantine Approximations, Lattices and flows on Homogeneous Spaces, A panorama of number theory or the view from Baker garden (2002), 280-310, Cambridge Univ. Press Zbl1036.11033
  19. C.A. Rogers, The signature of the errors of some simultaneous Diophantine approximations, Proc. London Math. Soc. 52 (1951), 186-190 Zbl0042.04601MR42465
  20. W.M. Schmidt, A metrical theorem in diophantine approximation, Canadian J. Math. 12 (1960), 619-631 Zbl0097.26205MR118711
  21. F. Schweiger, Multidimensional Continued Fractions, (2000), Oxford University Press Zbl0981.11029MR2121855
  22. V.G. Sprindžuk, Metric Theory of Diophantine Approximations, (1979), W.H. Winston Sons, Washington, C. D. Zbl0417.10044
  23. F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, 94 (1983), Springer-Verlag Zbl0516.58001MR722297

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.