A measure of transcendence for singular points on conics

Roy, Damien

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 2, pp. 361-369.

Un point d’une conique définie sur $\mathbb{Q}$ est dit singulier s’il est transcendant et admet de très bonnes approximations rationnelles, uniformément en termes de la hauteur. Les nombres extrémaux et les fractions continues sturmiennes sont les abscisses de tels points sur la parabole $y=x^2$. Nous établissons ici une mesure de transcendance de points singuliers sur les coniques définies sur $\mathbb{Q}$ qui, dans ces deux cas, améliore la mesure obtenue précédemment par Adamczewski et Bugeaud. L’outil principal est une version quantitative du théorème du sous-espace de Schmidt due à Evertse.

A singular point on a plane conic defined over $\mathbb{Q}$ is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola $y=x^2$. In this paper we provide a measure of transcendence for singular points on conics defined over $\mathbb{Q}$ which, in these two cases, improves on the measure obtained by Adamczewski and Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.

Classification : 11J82, 11J13, 11J87
Mots clés : Sturmian continued fractions, extremal numbers, transcendental numbers, measure of transcendence, uniform approximation, quantitative subspace theorem, minimal points, conics.

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