# 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

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topKraaikamp, 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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