Tong’s spectrum for Rosen continued fractions
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, p. 641-661

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.

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.

     author = {Kraaikamp, Cornelis and Schmidt, Thomas A. and Smeets, Ionica},
     title = {Tong's spectrum for Rosen continued fractions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {3},
     year = {2007},
     pages = {641-661},
     doi = {10.5802/jtnb.606},
     mrnumber = {2388792},
     zbl = {1173.11042},
     language = {en},
     url = {}
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.

[A] Adams, William W., On a relationship between the convergents of the nearest integer and regular continued fractions. Math. Comp. 33 (1979), no. 148, 1321–1331. | MR 537978 | Zbl 0426.10054

[B] Burger, E.B., Exploring the number jungle: a journey into Diophantine analysis. Student Mathematical Library, 8. American Mathematical Society, Providence, RI, 2000. | MR 1774066 | Zbl 0973.11001

[BJW] Bosma, W., Jager, H. and Wiedijk, F., Some metrical observations on the approximation by continued fractions. Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 281–299. | MR 718069 | Zbl 0519.10043

[BKS] Burton, R.M., Kraaikamp, C. and Schmidt, T.A., Natural extensions for the Rosen fractions. Trans. Amer. Math. Soc. 352 (1999), 1277–1298. | MR 1650073 | Zbl 0938.11036

[CF] Cusick, T.W. and Flahive, M.E., The Markoff and Lagrange spectra. Mathematical Surveys and Monographs, 30. American Mathematical Society, Providence, RI, 1989. | MR 1010419 | Zbl 0685.10023

[DK] Dajani, K. and Kraaikamp, C., Ergodic Theory of Numbers. The Carus Mathematical Monographs 29 (2002). | MR 1917322 | Zbl 1033.11040

[IK] Iosifescu, M. and Kraaikamp, C., Metrical Theory of Continued Fractions. Mathematics and its Applications, 547. Kluwer Academic Publishers, Dordrecht, 2002. | MR 1960327 | Zbl 1122.11047

[HK] Hartono, Y. and Kraaikamp, C., Tong’s spectrum for semi-regular continued fraction expansions. Far East J. Math. Sci. (FJMS) 13 (2004), no. 2, 137–165. | Zbl 1091.11029

[HS] Haas, A. and Series, C., Hurwitz constants and Diophantine approximation on Hecke groups. J. London Math. Soc. 34 (1986), 219–234. | MR 856507 | Zbl 0605.10018

[JK] Jager, H. and Kraaikamp, C., On the approximation by continued fractions. Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 3, 289–307. | MR 1020023 | Zbl 0695.10029

[KNS] Kraaikamp, C., Nakada, H. and Schmidt, T.A., On approximation by Rosen continued fractions. In preparation (2006).

[L] Legendre, A. M., Essai sur la théorie des nombres. Paris (1798).

[N1] Nakada, H., Continued fractions, geodesic flows and Ford circles. Algorithms, Fractals and Dynamics, (1995), 179–191. | MR 1402490 | Zbl 0868.30005

[N2] Nakada, H., On the Lenstra constant. Submitted (2007), eprint: arXiv: 0705.3756.

[R] Rosen, D., A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21 (1954), 549–563. | MR 65632 | Zbl 0056.30703

[T1] Tong, J.C., Approximation by nearest integer continued fractions. Math. Scand. 71 (1992), no. 2, 161–166. | MR 1212700 | Zbl 0787.11028

[T2] Tong, J.C., Approximation by nearest integer continued fractions. II. Math. Scand. 74 (1994), no. 1, 17–18. | MR 1277785 | Zbl 0812.11042