Rational approximation to real points on conics
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, p. 2331-2348

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.

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.

DOI : https://doi.org/10.5802/aif.2832
Classification:  11J13,  14H50
Keywords: algebraic curves, conics, real points, approximation by rational points, exponent of approximation, simultaneous approximation
@article{AIF_2013__63_6_2331_0,
     author = {Roy, Damien},
     title = {Rational approximation  to real points on conics},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     pages = {2331-2348},
     doi = {10.5802/aif.2832},
     mrnumber = {3237450},
     zbl = {06325436},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_6_2331_0}
}
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. http://www.numdam.org/item/AIF_2013__63_6_2331_0/

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