Markoff-Lagrange spectrum and extremal numbers

TL;DR

This paper explores extremal numbers related to the Markoff-Lagrange spectrum, establishing the existence of numbers with maximal Lagrange constant and linking their classification to solutions of Markoff's equation.

Contribution

It demonstrates the existence of extremal numbers with the largest Lagrange constant and connects their equivalence classes to Markoff's equation solutions.

Findings

Existence of extremal numbers with Lagrange constant 1/3.

A bijection between these numbers' classes and Markoff's solutions.

Stability of extremal numbers under GL_2(Z) transformations.

Abstract

Let gamma denote the golden ratio. H. Davenport and W. M.Schmidt showed in 1969 that, for each non-quadratic irrational real number xi, there exists a constant c>0 with the property that, for arbitrarily large values of X, the inequalities |x_0| \le X, |x_0*xi - x_1| \le cX^{-1/gamma}, |x_0*xi^2 - x_2| \le cX^{-1/gamma} admit no non-zero integer solution (x_0,x_1,x_2). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers xi such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X\ge 1. Such extremal numbers are transcendental and their set is stable under the action of GL_2(Z) by linear fractional transformations. In this paper, it is shown that there exists extremal numbers xi for which the Lagrange constant is 1/3, the largest possible value for a non-quadratic number, and that there is a natural bijection between the GL_2(Z)-equivalence classes of such numbers and the non-trivial solutions of Markoff's equation.