| Version 4 |
Version 3 |
| For any real $\xi$ which is not rational or quadratic irrational, there are infinitely many rational or real quadratic irrational $\alpha$ which satisfy |
For any real $\xi$ which is not rational or quadratic irrational, there are infinitely many rational or real quadratic irrational $\alpha$ which satisfy |
| \begin{displaymath} |
\begin{displaymath} |
| \mid \xi - \alpha \mid < C\cdot H(\alpha)^{-3}, |
\mid \xi - \alpha \mid < C\cdot H(\alpha)^{-3}, |
| \end{displaymath} |
\end{displaymath} |
| where |
where |
| \begin{displaymath} |
\begin{displaymath} |
| C = \left\{ |
C = \left\{ |
| \begin{array}{ll} |
\begin{array}{ll} |
| C_0, & \textrm{if} \mid\xi\mid < 1, \\ |
C_0, & \textrm{if} \mid\xi\mid < 1, \\ |
| C_0\cdot \xi^2, & \textrm{if} \mid\xi\mid >1. |
C_0\cdot \xi^2, & \textrm{if} \mid\xi\mid >1. |
| \end{array}\right |
\end{array}\right |
| \end{displaymath} |
\end{displaymath} |
| $C_0$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the \PMlinkescapetext{height} of $\alpha$. |
$C_0$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the \PMlinkescapetext{height} of $\alpha$. |
| \begin{thebibliography}{1} |
\begin{thebibliography}{1} |
|
\bibitem{DS} Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
|
\bibitem{DS} Davenport, H. & Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
|
| \end{thebibliography} |
\end{thebibliography} |
