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

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.

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.

Reçu le : 2012-04-20
Révisé le : 2013-04-06
Accepté le : 2014-03-13
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.869
Classification : 11J70,  11J13,  11H06
@article{JTNB_2014__26_2_307_0,
     author = {Christian Drouin},
     title = {A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {307--346},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.869},
     mrnumber = {3320482},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_307_0/}
}
Christian Drouin. A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 307-346. doi : 10.5802/jtnb.869. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_307_0/

[1] V.I. Arnold, Higher dimensional continued fractions. Regular and Chaotic Dynamics 3 (1998), n 3, 10–17. | MR 1704965 | Zbl 1044.11596

[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 638474 | Zbl 0471.10024

[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 1434478 | Zbl 0866.11041

[6] J.W.S. Cassels, An Introduction to diophantine approximation. Cambridge University Press, 1957. | MR 87708 | Zbl 0077.04801

[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 3284107

[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 67943 | Zbl 0064.04501

[11] H. Davenport, Simultaneous diophantine approximation. Proc. London Math. Soc. 2 (1952), 403–416. | MR 54657 | Zbl 0048.03204

[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 2394971 | Zbl 1180.11022

[14] J.F. Koksma, Diophantische Approximationen. Ergebnisse der Mathematik und ihrer Grenzgebiete 4 (1936), 409–571; and Chelsea Publishing Company, Amsterdam, 1982. | Zbl 0012.39602

[15] E Korkina, La périodicité des fractions continues multidimensionnelles, C. R. Acad. Sci. Paris t.319, Série I (1994), 777–780. | MR 1300940 | Zbl 0836.11023

[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 1244417 | Zbl 0809.52025

[17] J.C. Lagarias, Best simultaneous diophantine approximations I. Growth rates of best approximation denominators. Trans. Am. Math. Soc. 272 (1980), 545–554. | MR 662052 | Zbl 0495.10021

[18] J.C. Lagarias, Best simultaneous diophantine approximations II. Behavior of consecutive best approximations. Pacific J. Math. 102, n 1 (1982), 61–88. | MR 682045 | Zbl 0497.10025

[19] J.C. Lagarias, Geodesic multidimensional continued fractions, Proc. London Math. Soc, (3) (1994), 69, 231–244. | MR 1289860 | Zbl 0813.11040

[20] N.G. Moshchevitin, Continued fractions, multidimensional Diophantine approximations and applications. J. de Théorie des Nombres de Bordeaux 11 (1999), 425–438. | Numdam | MR 1745888 | Zbl 0987.11043

[21] W. Schmidt, Diophantine approximation. Lectures Notes in Mathematics 785, Springer, 1980. | MR 568710 | Zbl 0421.10019

[22] F. Schweiger, Multidimensional Continued Fractions Algorithms. Oxford University Press, 2000. | MR 2121855 | Zbl 0981.11029

[23] F. Schweiger, Was leisten mehrdimensionale Kettenbrüche. Mathematische Semesterberichte 53 (2006), 231–244. | MR 2251039 | Zbl 1171.11302