Best simultaneous diophantine approximations of some cubic algebraic numbers
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 403-414

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

Soit α un nombre algébrique réel de degré 3 dont les conjugués ne sont pas réels. Il existe une unité ζ de l’anneau des entiers de K=(α) pour laquelle il est possible de décrire l’ensemble de tous les vecteurs meilleurs approximations de θ=(ζ,ζ 2 ).

@article{JTNB_2002__14_2_403_0,
     author = {Chevallier, Nicolas},
     title = {Best simultaneous diophantine approximations of some cubic algebraic numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {403-414},
     zbl = {1071.11043},
     mrnumber = {2040684},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_403_0}
}
Chevallier, Nicolas. Best simultaneous diophantine approximations of some cubic algebraic numbers. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 403-414. http://www.numdam.org/item/JTNB_2002__14_2_403_0/

[1] W.W. Adams, Simultaneous diophantine Approximations and Cubic Irrationals. Pacific J. Math. 30 (1969), 1-14. | MR 245522 | Zbl 0182.37803

[2] W.W. Adams, Simultaneous Asymptotic diophantine Approximations to a Basis of a Real Cubic Field. J. Number Theory 1 (1969), 179-194. | MR 240055 | Zbl 0172.06501

[3] P. Bachmann, Zur Theory von Jacobi's Kettenbruch-Algorithmen, J. Reine Angew. Math. 75 (1873), 25-34. | JFM 04.0082.03

[4] L. Bernstein, The Jacobi-Perron algorithm-Its theory and applications, Lectures Notes in Mathematics 207, Springer-Verlag, 1971. | MR 285478 | Zbl 0213.05201

[5] A.J. Brentjes, Multi-dimensional continued fraction algorithms, Mathematics Center Tracts 155, Amsterdam, 1982. | MR 702520 | Zbl 0471.10024

[6] J.W.S. Cassels, An introduction to diophantine approximation. Cambridge University Press, 1965. | MR 87708

[7] N. Chekhova, P. Hubert, A. Messaoudi, Propriété combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Théor. Nombres Bordeaux 13 (2001), 371-394. | Numdam | MR 1879664 | Zbl 1038.37010

[8] N. Chevallier, Meilleures approximations d'un élément du tore T2 et géométrie de cet élément. Acta Arith. 78 (1996), 19-35. | MR 1424999 | Zbl 0863.11043

[9] E. Dubois, R. Paysant-Le Roux, Algorithme de Jacobi-Perron dans les extensions cubiques. C. R. Acad. Sci. Paris Sér. A 280 (1975), 183-186. | MR 360517 | Zbl 0297.12002

[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. | Zbl 0453.10035

[11] J.C. Lagarias, Best simultaneous diophantine approximation I. Growth rates of best approximations denominators. Trans. Amer. Math. Soc. 272 (1982), 545-554. | MR 662052 | Zbl 0495.10021

[12] H. Minkowski, Über periodish Approximationen Algebraischer Zalhen. Acta Math. 26 (1902), 333-351. | JFM 33.0216.02

[13] O. Perron, Grundlagen für eine Theorie des Jacobischen Kettenalgorithmus. Math. Ann. 64 (1907), 1-76. | JFM 38.0262.01 | MR 1511422

[14] G. Rauzy, Nombre algébrique et substitution. Bull. Soc. Math. France 110 (1982), 147-178. | Numdam | MR 667748 | Zbl 0522.10032