Roots of Markoff quadratic forms as strongly badly approximable numbers

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 $θ$, $θ'$ are $\pm$ equivalent if their sum or difference is an integer. Let $θ$ be irrational and put \[ϕ(θ) = \inf \{q \,\| q θ\| : q \in N \}. \] We will prove: If $ϕ(θ)> 1/3$, then $θ$ is $\pm$ equivalent to a root of $f_m (x,1) = 0$, where $f_m$ is a Markoff form. Conversely, if $θ$ is $\pm$ equivalent to a root of $f_m(x,1)=0$, then \[ϕ(θ) = m \| mθ\| = \frac{2}{3+\sqrt{9-4m^{-2}}} > 1/3. \]