Rational approximation to real points on conics

Roy, Damien

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2331-2348.

Un point $({\xi }_1,{\xi }_2)$ à coordonnées dans un sous-corps de $ℝ$ de degré de transcendance un sur $\mathbb{Q}$, avec $1,{\xi }_1,{\xi }_2$ linéairement indépendants sur $\mathbb{Q}$, peut admettre un exposant d’approximation uniforme par les éléments de ${\mathbb{Q}}^2$ 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 $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. Le but de cet article est de montrer que ce phénomène s’étend à toutes les coniques réelles définies sur $\mathbb{Q}$ 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 $1/\gamma \cong 0.618$ où $\gamma$ désigne le nombre d’or.

A point $({\xi }_1,{\xi }_2)$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $\mathbb{Q}$, with $1,{\xi }_1,{\xi }_2$ linearly independent over $\mathbb{Q}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ 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 $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\mathbb{Q}$, 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 $1/\gamma \cong 0.618$ where $\gamma$ denotes the golden ratio.

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

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