On the closedness of approximation spectra
Jouni Parkkonen1; Frédéric Paulin2
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, Volume 21 (2009) no. 3, pp. 703-712.

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].

Le spectre classique de Lagrange pour l’approximation des nombres réels par des rationnels, est fermé, par un théorème de Cusick. Plus généralement, nous montrons que de nombreux spectres d’approximation sont fermés, en utilisant des propriétés de pénétration du flot géodésique dans des voisinages de pointes de variétés à courbure strictement négative, et un résultat de Maucourant [Mau].

Received:
Published online:
DOI: 10.5802/jtnb.696
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 
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Jouni Parkkonen; Frédéric Paulin. On the closedness of approximation spectra. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 703-712. doi : 10.5802/jtnb.696. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.696/

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