# A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

Christian Drouin^{[1]}

- [1] 26 Avenue d’Yreye 40 510 SEIGNOSSE FRANCE

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 2, page 307-346
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topDrouin, Christian. "A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 307-346. <http://eudml.org/doc/275743>.

@article{Drouin2014,

abstract = {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.},

affiliation = {26 Avenue d’Yreye 40 510 SEIGNOSSE FRANCE},

author = {Drouin, Christian},

journal = {Journal de Théorie des Nombres de Bordeaux},

language = {eng},

month = {10},

number = {2},

pages = {307-346},

publisher = {Société Arithmétique de Bordeaux},

title = {A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties},

url = {http://eudml.org/doc/275743},

volume = {26},

year = {2014},

}

TY - JOUR

AU - Drouin, Christian

TI - A two-dimensional continued fraction algorithm with Lagrange and Dirichlet properties

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2014/10//

PB - Société Arithmétique de Bordeaux

VL - 26

IS - 2

SP - 307

EP - 346

AB - 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.

LA - eng

UR - http://eudml.org/doc/275743

ER -

## References

top- V.I. Arnold, Higher dimensional continued fractions. Regular and Chaotic Dynamics 3 (1998), n${}^{\circ}$3, 10–17. Zbl1044.11596MR1704965
- W. Bosma and I. Smeets, An algorithm for finding approximations with optimal Dirichlet quality (http://arxiv.org/abs/1001.4455). Submitted. Zbl06488038
- A.J. Brentjes, Multi-dimensional continued fraction algorithms. Mathematics Center Tracts 145, Mathematisch Centrum, Amsterdam, 1981. Zbl0471.10024MR638474
- K.M. Briggs, On the Furtwängler algorithm for simultaneous rational approximation. Exp. Math. (to be submitted), 2001.
- J.W.S. Cassels, An Introduction to the Geometry of Numbers. Springer. Zbl0866.11041MR1434478
- J.W.S. Cassels, An Introduction to diophantine approximation. Cambridge University Press, 1957. Zbl0077.04801MR87708
- N. Chevallier, Best Simultaneous Diophantine Approximations and Multidimensional Continued Fraction Expansions. Moscow J. of Combinatorics and Number Theory 3 (2013), n${}^{\circ}$1, 3–56. Zbl1305.11059MR3284107
- I.V.L. Clarkson, Approximation of Linear Forms by Lattice Points, with applications to signal processing. PhD thesis, Australian National University, 1997.
- V. Clarkson, J. Perkins, and I. Mareels, An algorithm for best approximation of a line by lattice points in three dimensions. Technical report, 1995. 3rd Conference on Computational Algebra and Number Theory (CANT 95). Formerly online at wwwcrasys.anu.edu.au/Projects/pulseTrain/Papers/CPM95.ps.gz.
- H. Davenport, On a theorem of Furtwängler. J. London Math. Soc. 30 (1955), 186–195. Zbl0064.04501MR67943
- H. Davenport, Simultaneous diophantine approximation. Proc. London Math. Soc. 2 (1952), 403–416. Zbl0048.03204MR54657
- Ph. Furtwängler, Über die simultane Approximation von Irrationalzahlen I and II. Math. Annalen 96 (1927), 169–175 and Math. Annalen 99 (1928), 71–83. Zbl54.0212.03
- O.N. German and E.L Lakshtanov, On a multidimensional generalization of Lagrange’s theorem on continued fractions. Izv. Math. 72:1 (2008), 47–61. Zbl1180.11022MR2394971
- J.F. Koksma, Diophantische Approximationen. Ergebnisse der Mathematik und ihrer Grenzgebiete 4 (1936), 409–571; and Chelsea Publishing Company, Amsterdam, 1982. Zbl0012.39602
- E Korkina,La périodicité des fractions continues multidimensionnelles, C. R. Acad. Sci. Paris t.319, Série I (1994), 777–780. Zbl0836.11023MR1300940
- G. Lachaud, Polyèdre d’Arnol’d et voile d’un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci. Paris, t. 317, Série I (1993), 711–716. Zbl0809.52025MR1244417
- J.C. Lagarias, Best simultaneous diophantine approximations I. Growth rates of best approximation denominators. Trans. Am. Math. Soc. 272 (1980), 545–554. Zbl0495.10021MR662052
- J.C. Lagarias, Best simultaneous diophantine approximations II. Behavior of consecutive best approximations. Pacific J. Math. 102, n${}^{\circ}$1 (1982), 61–88. Zbl0497.10025MR682045
- J.C. Lagarias,Geodesic multidimensional continued fractions, Proc. London Math. Soc, (3) (1994), 69, 231–244. Zbl0813.11040MR1289860
- N.G. Moshchevitin,Continued fractions, multidimensional Diophantine approximations and applications. J. de Théorie des Nombres de Bordeaux 11 (1999), 425–438. Zbl0987.11043MR1745888
- W. Schmidt, Diophantine approximation. Lectures Notes in Mathematics 785, Springer, 1980. Zbl0421.10019MR568710
- F. Schweiger, Multidimensional Continued Fractions Algorithms. Oxford University Press, 2000. Zbl0981.11029MR2121855
- F. Schweiger, Was leisten mehrdimensionale Kettenbrüche. Mathematische Semesterberichte 53 (2006), 231–244. Zbl1171.11302MR2251039

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.