Best simultaneous diophantine approximations of some cubic algebraic numbers

Nicolas Chevallier

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 403-414
  • ISSN: 1246-7405

Abstract

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Let α be a real algebraic number of degree 3 over ℚ whose conjugates are not real. There exists an unit ζ of the ring of integer of K = ℚ ( α ) for which it is possible to describe the set of all best approximation vectors of θ = ( ζ , ζ 2 ) .’

How to cite

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Chevallier, Nicolas. "Best simultaneous diophantine approximations of some cubic algebraic numbers." Journal de théorie des nombres de Bordeaux 14.2 (2002): 403-414. <http://eudml.org/doc/248920>.

@article{Chevallier2002,
abstract = {Let $\alpha $ be a real algebraic number of degree $3$ over $\mathbb \{Q\}$ whose conjugates are not real. There exists an unit $\zeta $ of the ring of integer of $K = \mathbb \{Q\}(\alpha )$ for which it is possible to describe the set of all best approximation vectors of $\theta = (\zeta , \zeta ^2)$.’},
author = {Chevallier, Nicolas},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {simultaneous diophantine approximation; real cubic numbers; Jacobi-Perron algorithm; continued fractions},
language = {eng},
number = {2},
pages = {403-414},
publisher = {Université Bordeaux I},
title = {Best simultaneous diophantine approximations of some cubic algebraic numbers},
url = {http://eudml.org/doc/248920},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Chevallier, Nicolas
TI - Best simultaneous diophantine approximations of some cubic algebraic numbers
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 403
EP - 414
AB - Let $\alpha $ be a real algebraic number of degree $3$ over $\mathbb {Q}$ whose conjugates are not real. There exists an unit $\zeta $ of the ring of integer of $K = \mathbb {Q}(\alpha )$ for which it is possible to describe the set of all best approximation vectors of $\theta = (\zeta , \zeta ^2)$.’
LA - eng
KW - simultaneous diophantine approximation; real cubic numbers; Jacobi-Perron algorithm; continued fractions
UR - http://eudml.org/doc/248920
ER -

References

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  2. [2] W.W. Adams, Simultaneous Asymptotic diophantine Approximations to a Basis of a Real Cubic Field. J. Number Theory1 (1969), 179-194. Zbl0172.06501MR240055
  3. [3] P. Bachmann, Zur Theory von Jacobi's Kettenbruch-Algorithmen, J. Reine Angew. Math. 75 (1873), 25-34. JFM04.0082.03
  4. [4] L. Bernstein, The Jacobi-Perron algorithm-Its theory and applications, Lectures Notes in Mathematics207, Springer-Verlag, 1971. Zbl0213.05201MR285478
  5. [5] A.J. Brentjes, Multi-dimensional continued fraction algorithms, Mathematics Center Tracts155, Amsterdam, 1982. Zbl0471.10024MR702520
  6. [6] J.W.S. Cassels, An introduction to diophantine approximation. Cambridge University Press, 1965. Zbl0077.04801MR87708
  7. [7] N. Chekhova, P. Hubert, A. Messaoudi, Propriété combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Théor. Nombres Bordeaux13 (2001), 371-394. Zbl1038.37010MR1879664
  8. [8] N. Chevallier, Meilleures approximations d'un élément du tore T2 et géométrie de cet élément. Acta Arith.78 (1996), 19-35. Zbl0863.11043MR1424999
  9. [9] E. Dubois, R. Paysant-Le Roux, Algorithme de Jacobi-Perron dans les extensions cubiques. C. R. Acad. Sci. Paris Sér. A280 (1975), 183-186. Zbl0297.12002MR360517
  10. [10] J.C. Lagarias, Some New results in simultaneous diophantine approximation. In Proc. of Queen's Number Theory Conference 1979 (P. Ribenboim, Ed.), Queen's Papers in Pure and Applied Math. No. 54 (1980), 453-574. Zbl0453.10035
  11. [11] J.C. Lagarias, Best simultaneous diophantine approximation I. Growth rates of best approximations denominators. Trans. Amer. Math. Soc.272 (1982), 545-554. Zbl0495.10021MR662052
  12. [12] H. Minkowski, Über periodish Approximationen Algebraischer Zalhen. Acta Math.26 (1902), 333-351. JFM33.0216.02
  13. [13] O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenalgorithmus. Math. Ann.64 (1907), 1-76. Zbl38.0262.01MR1511422JFM38.0262.01
  14. [14] G. Rauzy, Nombre algébrique et substitution. Bull. Soc. Math. France110 (1982), 147-178. Zbl0522.10032MR667748

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