Roots of Markoff quadratic forms as strongly badly approximable numbers

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