# 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

 $\mid\xi-\alpha\mid

where

 $C=\left\{\begin{array}[]{ll}C_{0},&\textrm{if}\mid\xi\mid<1,\\ C_{0}\cdot\xi^{2},&\textrm{if}\mid\xi\mid>1.\end{array}\right.$

$C_{0}$ 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

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

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]

## References

• DS Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
Title Davenport-Schmidt theorem DavenportSchmidtTheorem 2013-03-22 13:32:58 2013-03-22 13:32:58 Daume (40) Daume (40) 9 Daume (40) Theorem msc 11J68