Rational approximation to real points on conics

TL;DR

This paper investigates how certain algebraic points on all real conics over Q can be approximated more closely than the classical bounds suggest, revealing a universal approximation exponent related to the golden ratio.

Contribution

It extends the phenomenon of improved approximation exponents from parabola points to all real conics over Q, establishing a universal maximal exponent of approximately 0.618.

Findings

Points on all real conics over Q can have approximation exponents exceeding 1/2.

The maximal approximation exponent for such points is always 1/g, where g is the golden ratio.

This phenomenon is independent of the specific conic curve considered.

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.