Roots of Markoff quadratic forms as strongly badly approximable numbers

TL;DR

This paper characterizes strongly badly approximable numbers as roots of Markoff quadratic forms, establishing a precise link between their approximation properties and solutions to specific quadratic equations.

Contribution

It proves a bi-conditional relationship between the approximation constant \\phi(\\theta) and roots of Markoff forms, extending classical Diophantine approximation theory.

Findings

If \\phi(\\theta) > 1/3, then \\theta is \\pm equivalent to a root of a Markoff form.

Conversely, roots of Markoff forms have \\phi(\\theta) > 1/3, explicitly given by a formula.

The paper establishes a complete characterization of these numbers in terms of Markoff forms.

Abstract

For a real number $x$, $\| x\| = \min \{|x-p|: p\in Z\}$ is the distance of $x$ to the nearest integer. We say that two real numbers $\theta$, $\theta'$ are $\pm$ equivalent if their sum or difference is an integer. Let $\theta$ be irrational and put \[\phi(\theta) = \inf \{q \,\| q \theta\| : q \in N \}. \] We will prove: If $\phi(\theta)> 1/3$, then $\theta$ is $\pm$ equivalent to a root of $f_m (x,1) = 0$, where $f_m$ is a Markoff form. Conversely, if $\theta$ is $\pm$ equivalent to a root of $f_m(x,1)=0$, then \[\phi(\theta) = m \| m\theta \| = \frac{2}{3+\sqrt{9-4m^{-2}}} > 1/3. \]