In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients. We also obtain metrical results for large blocks of “bad” approximations.
Dans les années 90, J.C. Tong a donné une borne supérieure optimale pour le minimum de coefficients d’approximation consécutifs dans le cas des fractions continues à l’entier le plus proche. Nous généralisons ce type de résultat aux fractions continues de Rosen. Celles-ci constituent une famille infinie d’algorithmes de développement en fractions continues, où les quotients partiels sont certains entiers algébriques réels. Pour chacun de ces algorithmes nous déterminons la borne supérieure optimale de la valeur minimale des coefficients d’approximation pris en nombres consécutifs appropriés. Nous donnons aussi des résultats métriques pour des plages de “mauvaises” approximations successives de grande longueur.
@article{JTNB_2007__19_3_641_0, author = {Kraaikamp, Cornelis and Schmidt, Thomas A. and Smeets, Ionica}, title = {Tong{\textquoteright}s spectrum for {Rosen} continued fractions}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {641--661}, publisher = {Universit\'e Bordeaux 1}, volume = {19}, number = {3}, year = {2007}, doi = {10.5802/jtnb.606}, zbl = {1173.11042}, mrnumber = {2388792}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.606/} }
TY - JOUR AU - Kraaikamp, Cornelis AU - Schmidt, Thomas A. AU - Smeets, Ionica TI - Tong’s spectrum for Rosen continued fractions JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 641 EP - 661 VL - 19 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.606/ DO - 10.5802/jtnb.606 LA - en ID - JTNB_2007__19_3_641_0 ER -
%0 Journal Article %A Kraaikamp, Cornelis %A Schmidt, Thomas A. %A Smeets, Ionica %T Tong’s spectrum for Rosen continued fractions %J Journal de théorie des nombres de Bordeaux %D 2007 %P 641-661 %V 19 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.606/ %R 10.5802/jtnb.606 %G en %F JTNB_2007__19_3_641_0
Kraaikamp, Cornelis; Schmidt, Thomas A.; Smeets, Ionica. Tong’s spectrum for Rosen continued fractions. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 641-661. doi : 10.5802/jtnb.606. http://archive.numdam.org/articles/10.5802/jtnb.606/
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