no. 6
Rational approximation to real points on conics
Roy, Damien1
1Université d’Ottawa Département de Mathématiques 585 King Edward Ottawa, Ontario K1N 6N5 (Canada)
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2331-2348

A point (ξ 1 ,ξ 2 ) with coordinates in a subfield of ℝ of transcendence degree one over ℚ, with 1,ξ 1 ,ξ 2 linearly independent over ℚ, may have a uniform exponent of approximation by elements of ℚ 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∈ℝ}. The goal of this paper is to show that this phenomenon extends to all real conics defined over ℚ, 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/γ≅0.618 where γ denotes the golden ratio.

Un point (ξ 1 ,ξ 2 ) à coordonnées dans un sous-corps de ℝ de degré de transcendance un sur ℚ, avec 1,ξ 1 ,ξ 2 linéairement indépendants sur ℚ, peut admettre un exposant d’approximation uniforme par les éléments de ℚ 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∈ℝ}. Le but de cet article est de montrer que ce phénomène s’étend à toutes les coniques réelles définies sur ℚ 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/γ≅0.618 où γ désigne le nombre d’or.

DOI: 10.5802/aif.2832
Classification: 11J13, 14H50
Keywords: algebraic curves, conics, real points, approximation by rational points, exponent of approximation, simultaneous approximation
Mots-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) 
Roy, Damien. Rational approximation  to real points on conics. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2331-2348. doi: 10.5802/aif.2832
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