Davenport-Schmidt theorem

For any real $\xi$ which is not rational or quadratic irrational, there are infinitely many rational or real quadratic irrational $\alpha$ which satisfy

where

$C_0$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the of $\alpha$.[DS]
The of the rational or quadratic irrational number $\alpha$ is

$$H(\alpha )=\max (|x|,|y|,|z|)$$

where $x$,$y$,$z$ are from the unique equation

$$x{\alpha }^2+y\alpha +z=0$$

such that $x$,$y$,$z$ are not all zero relatively prime integral coefficients.[DS]