On the closedness of approximation spectra

Jouni Parkkonen[1]; Frédéric Paulin[2]

  • [1] Department of Mathematics and Statistics P.O. Box 35 40014 University of Jyväskylä, FINLAND
  • [2] Département de Mathématique et Applications, UMR 8553 CNRS École Normale Supérieure 45 rue d’Ulm 75230 PARIS Cedex 05, FRANCE

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 703-712
  • ISSN: 1246-7405

Abstract

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Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].

How to cite

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Parkkonen, Jouni, and Paulin, Frédéric. "On the closedness of approximation spectra." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 703-712. <http://eudml.org/doc/10906>.

@article{Parkkonen2009,
abstract = {Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].},
affiliation = {Department of Mathematics and Statistics P.O. Box 35 40014 University of Jyväskylä, FINLAND; Département de Mathématique et Applications, UMR 8553 CNRS École Normale Supérieure 45 rue d’Ulm 75230 PARIS Cedex 05, FRANCE},
author = {Parkkonen, Jouni, Paulin, Frédéric},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Diophantine approximation; Lagrange spectrum; quadratic imaginary rings; the quaternions; Heisenberg group},
language = {eng},
number = {3},
pages = {703-712},
publisher = {Université Bordeaux 1},
title = {On the closedness of approximation spectra},
url = {http://eudml.org/doc/10906},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Parkkonen, Jouni
AU - Paulin, Frédéric
TI - On the closedness of approximation spectra
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 703
EP - 712
AB - Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].
LA - eng
KW - Diophantine approximation; Lagrange spectrum; quadratic imaginary rings; the quaternions; Heisenberg group
UR - http://eudml.org/doc/10906
ER -

References

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  10. S. Hersonsky, F. Paulin, Diophantine Approximation on Negatively Curved Manifolds and in the Heisenberg Group. In “Rigidity in dynamics and geometry” (Cambridge, 2000), M. Burger, A. Iozzi eds, Springer Verlag (2002), 203–226. Zbl1064.11057
  11. R. Kellerhals, Quaternions and some global properties of hyperbolic 5 -manifolds, Canad. J. Math. 55 (2003) 1080–1099. Zbl1054.57019MR2005283
  12. F. Maucourant, Sur les spectres de Lagrange et de Markoff des corps imaginaires quadratiques. Erg. Theo. Dyn. Sys. 23 (2003), 193–205. Zbl1049.11068MR1971202
  13. J. Parkkonen, F. Paulin, Appendix: Diophantine Approximation on Hyperbolic Surfaces. In “Rigidity in dynamics and geometry” (Cambridge, 2000), M. Burger, A. Iozzi eds, Springer Verlag (2002), 227–236. Zbl1064.11058MR1919403
  14. J. Parkkonen, F. Paulin, Sur les rayons de Hall en approximation diophantienne. Comptes Rendus Math. 344 (2007), 611–614. Zbl1151.11030MR2334070
  15. J. Parkkonen, F. Paulin, Prescribing the behaviour of geodesics in negative curvature. Geometry & Topology 14 (2010), 277–392. Zbl1191.53026
  16. M. F. Vigneras, Arithmétique des algèbres de quaternions. Lect. Notes 800, Springer Verlag, 1980. Zbl0422.12008MR580949

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