Roots of Markoff quadratic forms as strongly badly approximable numbers

Abstract

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

15 pages