Exponents of Diophantine Approximation and Sturmian Continued Fractions

Yann Bugeaud[1]; Michel Laurent

  • [1] Université Louis Pasteur, U. F. R. de mathématiques, 7 rue René Descartes, 67084 STRASBOURG (France), Institut de Mathématiques de Luminy, U.P.R. 9016, case 907, 163 avenue de Luminy, 13288 MARSEILLE CEDEX 9 (France)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 3, page 773-804
  • ISSN: 0373-0956

Abstract

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Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n ( ξ ) and w n * ( ξ ) defined by Mahler and Koksma. We calculate their six values when n = 2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

How to cite

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Bugeaud, Yann, and Laurent, Michel. "Exponents of Diophantine Approximation and Sturmian Continued Fractions." Annales de l’institut Fourier 55.3 (2005): 773-804. <http://eudml.org/doc/116208>.

@article{Bugeaud2005,
abstract = {Let $\xi $ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n(\xi )$ and $w_n^*(\xi )$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi $ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi $ by quadratic surds.},
affiliation = {Université Louis Pasteur, U. F. R. de mathématiques, 7 rue René Descartes, 67084 STRASBOURG (France), Institut de Mathématiques de Luminy, U.P.R. 9016, case 907, 163 avenue de Luminy, 13288 MARSEILLE CEDEX 9 (France)},
author = {Bugeaud, Yann, Laurent, Michel},
journal = {Annales de l’institut Fourier},
keywords = {Diophantine approximation; Sturmian sequence; simultaneous approximation; transcendence measure},
language = {eng},
number = {3},
pages = {773-804},
publisher = {Association des Annales de l'Institut Fourier},
title = {Exponents of Diophantine Approximation and Sturmian Continued Fractions},
url = {http://eudml.org/doc/116208},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Bugeaud, Yann
AU - Laurent, Michel
TI - Exponents of Diophantine Approximation and Sturmian Continued Fractions
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 3
SP - 773
EP - 804
AB - Let $\xi $ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n(\xi )$ and $w_n^*(\xi )$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi $ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi $ by quadratic surds.
LA - eng
KW - Diophantine approximation; Sturmian sequence; simultaneous approximation; transcendence measure
UR - http://eudml.org/doc/116208
ER -

References

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