Rational approximation to real points on conics
[Approximation rationnelle de points réels sur les coniques]
Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2331-2348.

Un point (ξ 1 ,ξ 2 ) à coordonnées dans un sous-corps de de degré de transcendance un sur , avec 1,ξ 1 ,ξ 2 linéairement indépendants sur , peut admettre un exposant d’approximation uniforme par les éléments de 2 qui soit strictement plus grand que la borne inférieure 1/2 que garantit le principe des tiroirs de Dirichlet. Ce fait inattendu est apparu, en lien avec des travaux de Davenport et Schmidt, pour les points de la parabole {(ξ,ξ 2 );ξ}. Le but de cet article est de montrer que ce phénomène s’étend à toutes les coniques réelles définies sur et que le plus grand exposant d’approximation atteint par les points de ces courbes, sujets à la condition d’indépendance linéaire mentionnée plus tôt, est toujours le même, indépendamment de la courbe, à savoir 1/γ0.618γ désigne le nombre d’or.

A point (ξ 1 ,ξ 2 ) with coordinates in a subfield of of transcendence degree one over , with 1,ξ 1 ,ξ 2 linearly independent over , may have a uniform exponent of approximation by elements of 2 that is strictly larger than the lower bound 1/2 given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola {(ξ,ξ 2 );ξ}. The goal of this paper is to show that this phenomenon extends to all real conics defined over , and that the largest exponent of approximation achieved by points of these curves satisfying the above condition of linear independence is always the same, independently of the curve, namely 1/γ0.618 where γ denotes the golden ratio.

DOI : 10.5802/aif.2832
Classification : 11J13, 14H50
Keywords: algebraic curves, conics, real points, approximation by rational points, exponent of approximation, simultaneous approximation
Mot clés : courbes algébriques, coniques, points réels, approximation par des points rationnels, exposant d’approximation, approximation simultanée
Roy, Damien 1

1 Université d’Ottawa Département de Mathématiques 585 King Edward Ottawa, Ontario K1N 6N5 (Canada)
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Roy, Damien. Rational approximation  to real points on conics. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2331-2348. doi : 10.5802/aif.2832. http://www.numdam.org/articles/10.5802/aif.2832/

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