A measure of transcendence for singular points on conics

TL;DR

This paper develops a new measure of transcendence for singular points on conics over rationals, improving previous bounds for special cases like extremal numbers and Sturmian continued fractions.

Contribution

It introduces a quantitative measure of transcendence for singular points on conics, enhancing existing results using a refined Schmidt subspace theorem.

Findings

Improved transcendence measure for extremal numbers

Enhanced bounds for Sturmian continued fractions

Application of Evertse's quantitative Schmidt subspace theorem

Abstract

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 et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.