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.

[1] Bel, P. Approximation simultanée d’un nombre $v$-adique et de son carré par des nombres algébriques, J. Number Theory (to appear)

[2] Bugeaud, Yann; Laurent, Michel Exponents of Diophantine approximation and Sturmian continued fractions, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 3, pp. 773-804 | DOI | Numdam | MR | Zbl

[3] Davenport, H.; Schmidt, Wolfgang M. Approximation to real numbers by algebraic integers, Acta Arith., Volume 15 (1968/1969), pp. 393-416 | MR | Zbl

[4] Kleinbock, D. Extremal subspaces and their submanifolds, Geom. Funct. Anal., Volume 13 (2003) no. 2, pp. 437-466 | DOI | MR | Zbl

[5] Laurent, Michel Simultaneous rational approximation to the successive powers of a real number, Indag. Math. (N.S.), Volume 14 (2003) no. 1, pp. 45-53 | DOI | MR | Zbl

[6] Lozier, S.; Roy, D. Simultaneous approximation to a real number and to its cube, Acta Arith. (to appear)

[7] Roy, Damien Approximation simultanée d’un nombre et de son carré, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 1, pp. 1-6 | DOI | MR | Zbl

[8] Roy, Damien Approximation to real numbers by cubic algebraic integers. II, Ann. of Math. (2), Volume 158 (2003) no. 3, pp. 1081-1087 | DOI | MR | Zbl

[9] Roy, Damien Approximation to real numbers by cubic algebraic integers. I, Proc. London Math. Soc. (3), Volume 88 (2004) no. 1, pp. 42-62 | DOI | MR | Zbl

[10] Roy, Damien On two exponents of approximation related to a real number and its square, Canad. J. Math., Volume 59 (2007) no. 1, pp. 211-224 | DOI | MR | Zbl

[11] Roy, Damien On simultaneous rational approximations to a real number, its square, and its cube, Acta Arith., Volume 133 (2008) no. 2, pp. 185-197 | DOI | MR | Zbl

[12] Schmidt, Wolfgang M. Diophantine approximation, Lecture Notes in Mathematics, 785, Springer, Berlin, 1980, pp. x+299 | MR | Zbl

[13] Zelo, Dmitrij Simultaneous approximation to real and p-adic numbers, ProQuest LLC, Ann Arbor, MI, 2009, pp. 147 Thesis (Ph.D.)–University of Ottawa (Canada) | MR