Geodesic
Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction
Čebyševskij sbornik, Tome 11 (2010) no. 1, pp. 68-73 
A. D. Bruno. Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction. Čebyševskij sbornik, Tome 11 (2010) no. 1, pp. 68-73. http://geodesic.mathdoc.fr/item/CHEB_2010_11_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let in a three-dimensional real space two forms be given: a linear form and a quadratic one which is a product of two complex conjugate linear forms. Their root sets are a plane and a straight line correspondingly. We assume that the line does not lie in the plane. Voronoi (1896) and author (2006) proposed two different algorithms for computation of integer points giving the best approximations to roots of these two forms. The both algorithms are one-way: the Voronoi algorithms is directed to the plane and the authors algorithms is directed to the line. Here we propose an algorithm, which works in both directions. We give also a survey of results on such approach to simultaneous Diophantine approximations in any dimensions.

[1] G. F. Voronoi, On Generalization of the Algorithm of Continued Fractions, Warszawa University, 1896; Collected Works in 3 Volumes, v. 1, Izdat. Akad. Nauk USSR, Kiev, 1952 (in Russian) | MR

[2] Doklady Mathematics, 74:2 (2006), 628–632 | MR | MR | Zbl

[3] Doklady Mathematics, 80:3 (2009), 887–890 | MR | Zbl

[4] Doklady Mathematics, 71:3 (2005), 396–400 | MR

[5] Doklady Mathematics, 71:3 (2005), 446–450 | MR

[6] A. D. Bruno, “Generalizations of continued fraction”, Chebyshevskii sbornik, 7:3 (2006), 4–71 | MR

[7] Doklady Mathematics, 82:1 (2010)

[8] F. Klein, “Über eine geometrische Auffassung der gewöhnlichen Kettenbruchentwicklung”, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl., 1895, no. 3, 357–359 | Zbl

[9] H. Minkowski, “Généralisation de le théorie des fractions continues”, Ann. Sci. Ec. Norm. Super. ser. III, 13 (1896), 41–60 ; Gesamm. Abh., I, 278–292 | MR | Zbl

[10] B. F. Skubenko, “Minimum of a decomposable cubic form of three variables”, J. Sov. Math., 53:3 (1991), 302–321 | MR | Zbl

[11] V. I. Arnold, “Higher dimensional continued fractions”, Regular and Chaotic Dynamics, 3:3 (1998), 10–17 | MR | Zbl

[12] Math. Notes, 56:3–4 (1994), 994–1007 | MR | MR

[13] G. Lachaud, “Polyèdre d'Arnol'd et voile d'un cône simplicial: analogues du théorème de Lagrange”, C. R. Acad. Sci. Ser. 1, 317 (1993), 711–716 | MR | Zbl

[14] Mathem. Notes, 61:3 (1997), 278–286 | MR

[15] V. I. Parusnikov, “Klein polyhedra for complete decomposable forms”, Number theory. Diophantine, Computational and Algebraic Aspects, eds. K. Győry, A. Pethő, V. T. Sós, De Gruyter, Berlin–New York, 1998, 453–463 | MR | Zbl

[16] Math. Notes, 67:1 (2000), 87–102 | MR | Zbl

[17] Math. Notes, 77:4 (2005), 523–538 (in English) | MR | Zbl

[18] J. C. Lagarias, “Geodesic multidimensional continued fractions”, Proc. London Math. Soc. (3), 69 (1994), 464–488 | MR | Zbl