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Davenport-Schmidt theorem (Theorem)

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

$\displaystyle \mid \xi - \alpha \mid < C\cdot H(\alpha)^{-3}, $
where
\begin{displaymath} C = \left\{ \begin{array}{ll} C_0, & \textrm{if} \mid\xi\mid... ...0\cdot \xi^2, & \textrm{if} \mid\xi\mid >1. \end{array}\right. \end{displaymath}
$ C_0$ is any fixed number greater than $ \frac{160}{9}$ and $ H(\alpha )$ is the height of $ \alpha$.[DS]
The height of the rational or quadratic irrational number $ \alpha$ is
$\displaystyle H(\alpha)=\operatorname{max}(\vert x\vert,\vert y\vert,\vert z\vert)$
where $ x$,$ y$,$ z$ are from the unique equation
$\displaystyle x\alpha^2+y\alpha+z=0$
such that $ x$,$ y$,$ z$ are not all zero relatively prime integral coefficients.[DS]

Bibliography

DS
Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.



"Davenport-Schmidt theorem" is owned by Daume.
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Cross-references: coefficients, integral, relatively prime, equation, fixed, irrational, rational, real

This is version 6 of Davenport-Schmidt theorem, born on 2003-04-04, modified 2004-06-10.
Object id is 4151, canonical name is DavenportSchmidt.
Accessed 1637 times total.

Classification:
AMS MSC: 11J68 (Number theory :: Diophantine approximation, transcendental number theory :: Approximation to algebraic numbers)
Pending Errata and Addenda
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