Measures of algebraic approximation to Markoff extremal numbers

TL;DR

This paper investigates algebraic approximation properties of certain transcendental numbers related to Markoff extremal numbers, revealing optimal bounds for degrees 3 to 5 but not for degree 6.

Contribution

It demonstrates the optimality of approximation estimates for specific transcendental numbers for degrees 3 to 5, but not for degree 6, extending previous work on algebraic approximation.

Findings

Optimal approximation bounds for degrees 3, 4, 5

Failure of bounds for degree 6

Extension of Davenport and Schmidt's framework

Abstract

Let xi be a real number which is neither rational nor quadratic over Q. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number theta, there exist a constant c>0 and infinitely many non-zero polynomials P in Z[T] of degree at most 2 such that |theta-P(xi)| < c |P|^{-gamma} where gamma=(1+sqrt{5})/2 denotes for the golden ratio and where the norm |P| of P stands for the largest absolute value of its coefficients. In the present paper, we show conversely that there exists a class of transcendental numbers xi for which the above estimates are optimal up to the value of the constant c when one takes theta=R(xi) for a polynomial R in Z[T] of degree d = 3, 4 or 5 but curiously not for degree d=6, even with theta = 2 xi^6.