Tong’s spectrum for Rosen continued fractions

Kraaikamp, Cornelis; Schmidt, Thomas A.; Smeets, Ionica

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 641-661.

Dans les années 90, J.C. Tong a donné une borne supérieure optimale pour le minimum de $k$ 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.

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of $k$ 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.

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