Continued fractions and Parametric geometry of numbers
Aminata Keita1
1 Département de Mathématiques Université d’Ottawa 585 King Edward Ottawa, Ontario K1N 6N5, Canada
Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 129-135.

Recently, W. M. Schmidt and L. Summerer developed a new theory called Parametric Geometry of Numbers which approximates the behaviour of the successive minima of a family of convex bodies in ℝ n related to the problem of simultaneous rational approximation to given real numbers. In the case of one number, we show that the qualitative behaviour of the minima reflects the continued fraction expansion of the smallest distance from this number to an integer.

W. M. Schmidt et L. Summerer ont récemment proposé une théorie qui décrit approximativement le comportement des minima successifs d’une famille de corps convexes à un paramètre, celle qui intervient dans les problèmes d’approximation rationnelle simultanée de plusieurs nombres. Dans le cas d’un seul nombre, nous montrons que le comportement qualitatif exact des minima reflète le développement en fraction continue de la plus petite distance de ce nombre à un entier.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.971
Classification: 11J04, 11J82
Keywords: Convex body, Successive minima, Combined graph, Continue fractions

Aminata Keita 1

1 Département de Mathématiques Université d’Ottawa 585 King Edward Ottawa, Ontario K1N 6N5, Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Aminata Keita. Continued fractions and  Parametric geometry of numbers. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 129-135. doi : 10.5802/jtnb.971. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.971/

[1] D. Roy On Schmidt and Summerer parametric geometry of numbers (Annals of Mathematics, to appear)

[2] Wolfgang M. Schmidt Diophantine approximation, Lecture Notes in Mathematics, 785, Springer-Verlag, 1980, x+299 pages

[3] Wolfgang M. Schmidt; Leonhard Summerer Parametric geometry of numbers and applications, Acta Arith., Volume 140 (2009) no. 1, pp. 67-91 | DOI

[4] Wolfgang M. Schmidt; Leonhard Summerer Diophantine approximation and parametric geometry of numbers, Monatsh. Math., Volume 169 (2013) no. 1, pp. 51-104 | DOI

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