Tong’s spectrum for Rosen continued fractions
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 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.

Received:
Published online:
DOI: 10.5802/jtnb.606
Cornelis Kraaikamp 1; Thomas A. Schmidt 2; Ionica Smeets 3

1 EWI, Delft University of Technology, Mekelweg 4, 2628 CD Delft the Netherlands
2 Oregon State University Corvallis, OR 97331 USA
3 Mathematical Institute Leiden University Niels Bohrweg 1, 2333 CA Leiden the Netherlands
@article{JTNB_2007__19_3_641_0,
     author = {Cornelis Kraaikamp and Thomas A. Schmidt and Ionica Smeets},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.606/}
}
TY  - JOUR
TI  - Tong’s spectrum for Rosen continued fractions
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2007
DA  - 2007///
SP  - 641
EP  - 661
VL  - 19
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.606/
UR  - https://zbmath.org/?q=an%3A1173.11042
UR  - https://www.ams.org/mathscinet-getitem?mr=2388792
UR  - https://doi.org/10.5802/jtnb.606
DO  - 10.5802/jtnb.606
LA  - en
ID  - JTNB_2007__19_3_641_0
ER  - 
%0 Journal Article
%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 https://doi.org/10.5802/jtnb.606
%R 10.5802/jtnb.606
%G en
%F JTNB_2007__19_3_641_0
Cornelis Kraaikamp; Thomas A. Schmidt; Ionica Smeets. 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. https://jtnb.centre-mersenne.org/articles/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

[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

[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

[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

Cited by Sources: