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].
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DOI: 10.5802/jtnb.696
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@article{JTNB_2009__21_3_703_0, author = {Jouni Parkkonen and Fr\'ed\'eric Paulin}, title = {On the closedness of approximation spectra}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {703--712}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {3}, year = {2009}, doi = {10.5802/jtnb.696}, zbl = {1205.11083}, mrnumber = {2605541}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.696/} }
TY - JOUR TI - On the closedness of approximation spectra JO - Journal de Théorie des Nombres de Bordeaux PY - 2009 DA - 2009/// SP - 703 EP - 712 VL - 21 IS - 3 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.696/ UR - https://zbmath.org/?q=an%3A1205.11083 UR - https://www.ams.org/mathscinet-getitem?mr=2605541 UR - https://doi.org/10.5802/jtnb.696 DO - 10.5802/jtnb.696 LA - en ID - JTNB_2009__21_3_703_0 ER -
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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