Rational approximation to real points on conics

Abstract

A point (x1, x2) with coordinates in a subfield of R of transcendence degree one over Q, with 1, x1, x2 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 {(x, x^2) ; x in 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/g \approx 0.618 where g denotes the golden ratio.