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<C\cdot H(\alpha)^{-3},

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
Canonical name	DavenportSchmidtTheorem
Date of creation	2013-03-22 13:32:58
Last modified on	2013-03-22 13:32:58
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	9
Author	Daume (40)
Entry type	Theorem
Classification	msc 11J68