A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 307-346.

A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued fractions algorithms.

On démontre dans cet article un Théorème de Lagrange, pour un certain algorithme de fraction continue en dimension 2, dont la définition géométrique est très naturelle. Des propriétés type Dirichlet sont aussi obtenues pour la convergence de cet algorithme. Ces propriétés proviennent de caractéristiques géométriques de l’algorithme. Les relations entre ces différentes propriétés sont étudiées. En lien avec l’algorithme présenté, sont rapidement évoqués les travaux de divers auteurs dans le domaine des fractions continues multidimensionnelles.

DOI: 10.5802/jtnb.869
Classification: 11J70, 11J13, 11H06
Drouin, Christian 1

1 26 Avenue d’Yreye 40 510 SEIGNOSSE FRANCE
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Drouin, Christian. A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 307-346. doi : 10.5802/jtnb.869. http://archive.numdam.org/articles/10.5802/jtnb.869/

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

[2] W. Bosma and I. Smeets, An algorithm for finding approximations with optimal Dirichlet quality (http://arxiv.org/abs/1001.4455). Submitted.

[3] A.J. Brentjes, Multi-dimensional continued fraction algorithms. Mathematics Center Tracts 145, Mathematisch Centrum, Amsterdam, 1981. | MR | Zbl

[4] K.M. Briggs, On the Furtwängler algorithm for simultaneous rational approximation. Exp. Math. (to be submitted), 2001.

[5] J.W.S. Cassels, An Introduction to the Geometry of Numbers. Springer. | MR | Zbl

[6] J.W.S. Cassels, An Introduction to diophantine approximation. Cambridge University Press, 1957. | MR | Zbl

[7] N. Chevallier, Best Simultaneous Diophantine Approximations and Multidimensional Continued Fraction Expansions. Moscow J. of Combinatorics and Number Theory 3 (2013), n 1, 3–56. | MR

[8] I.V.L. Clarkson, Approximation of Linear Forms by Lattice Points, with applications to signal processing. PhD thesis, Australian National University, 1997.

[9] V. Clarkson, J. Perkins, and I. Mareels, An algorithm for best approximation of a line by lattice points in three dimensions. Technical report, 1995. 3rd Conference on Computational Algebra and Number Theory (CANT 95). Formerly online at wwwcrasys.anu.edu.au/Projects/pulseTrain/Papers/CPM95.ps.gz.

[10] H. Davenport, On a theorem of Furtwängler. J. London Math. Soc. 30 (1955), 186–195. | MR | Zbl

[11] H. Davenport, Simultaneous diophantine approximation. Proc. London Math. Soc. 2 (1952), 403–416. | MR | Zbl

[12] Ph. Furtwängler, Über die simultane Approximation von Irrationalzahlen I and II. Math. Annalen 96 (1927), 169–175 and Math. Annalen 99 (1928), 71–83.

[13] O.N. German and E.L Lakshtanov, On a multidimensional generalization of Lagrange’s theorem on continued fractions. Izv. Math. 72:1 (2008), 47–61. | MR | Zbl

[14] J.F. Koksma, Diophantische Approximationen. Ergebnisse der Mathematik und ihrer Grenzgebiete 4 (1936), 409–571; and Chelsea Publishing Company, Amsterdam, 1982. | Zbl

[15] E Korkina, La périodicité des fractions continues multidimensionnelles, C. R. Acad. Sci. Paris t.319, Série I (1994), 777–780. | MR | Zbl

[16] G. Lachaud, Polyèdre d’Arnol’d et voile d’un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci. Paris, t. 317, Série I (1993), 711–716. | MR | Zbl

[17] J.C. Lagarias, Best simultaneous diophantine approximations I. Growth rates of best approximation denominators. Trans. Am. Math. Soc. 272 (1980), 545–554. | MR | Zbl

[18] J.C. Lagarias, Best simultaneous diophantine approximations II. Behavior of consecutive best approximations. Pacific J. Math. 102, n 1 (1982), 61–88. | MR | Zbl

[19] J.C. Lagarias, Geodesic multidimensional continued fractions, Proc. London Math. Soc, (3) (1994), 69, 231–244. | MR | Zbl

[20] N.G. Moshchevitin, Continued fractions, multidimensional Diophantine approximations and applications. J. de Théorie des Nombres de Bordeaux 11 (1999), 425–438. | Numdam | MR | Zbl

[21] W. Schmidt, Diophantine approximation. Lectures Notes in Mathematics 785, Springer, 1980. | MR | Zbl

[22] F. Schweiger, Multidimensional Continued Fractions Algorithms. Oxford University Press, 2000. | MR | Zbl

[23] F. Schweiger, Was leisten mehrdimensionale Kettenbrüche. Mathematische Semesterberichte 53 (2006), 231–244. | MR | Zbl

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