Tong’s spectrum for Rosen continued fractions

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 641-661
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Kraaikamp, Cornelis, Schmidt, Thomas A., and Smeets, Ionica. "Tong’s spectrum for Rosen continued fractions." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 641-661. <http://eudml.org/doc/249964>.

@article{Kraaikamp2007,
abstract = {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.},
affiliation = {EWI, Delft University of Technology, Mekelweg 4, 2628 CD Delft the Netherlands; Oregon State University Corvallis, OR 97331 USA; Mathematical Institute Leiden University Niels Bohrweg 1, 2333 CA Leiden the Netherlands},
author = {Kraaikamp, Cornelis, Schmidt, Thomas A., Smeets, Ionica},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {641-661},
publisher = {Université Bordeaux 1},
title = {Tong’s spectrum for Rosen continued fractions},
url = {http://eudml.org/doc/249964},
volume = {19},
year = {2007},
}

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
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 641
EP - 661
AB - 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.
LA - eng
UR - http://eudml.org/doc/249964
ER -

References

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  11. Kraaikamp, C., Nakada, H. and Schmidt, T.A., On approximation by Rosen continued fractions. In preparation (2006). 
  12. Legendre, A. M., Essai sur la théorie des nombres. Paris (1798). 
  13. Nakada, H., Continued fractions, geodesic flows and Ford circles. Algorithms, Fractals and Dynamics, (1995), 179–191. Zbl0868.30005MR1402490
  14. Nakada, H., On the Lenstra constant. Submitted (2007), eprint: arXiv: 0705.3756. Zbl1205.11091
  15. Rosen, D., A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21 (1954), 549–563. Zbl0056.30703MR65632
  16. Tong, J.C., Approximation by nearest integer continued fractions. Math. Scand. 71 (1992), no. 2, 161–166. Zbl0787.11028MR1212700
  17. Tong, J.C., Approximation by nearest integer continued fractions. II. Math. Scand. 74 (1994), no. 1, 17–18. Zbl0812.11042MR1277785

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