Davenport-Schmidt theorem
DavenportSchmidt
2003-04-04 20:04:23
2004-06-10 22:11:51
Theorem
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For any real $\xi$ which is not rational or quadratic irrational, there are infinitely many rational or real quadratic irrational $\alpha$ which satisfy
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\mid \xi - \alpha \mid < C\cdot H(\alpha)^{-3},
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where
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C = \left\{
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C_0, & \textrm{if} \mid\xi\mid < 1, \\
C_0\cdot \xi^2, & \textrm{if} \mid\xi\mid >1.
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$C_0$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the \PMlinkescapetext{height} of $\alpha$.\cite{DS}\\
The \emph{\PMlinkescapetext{height} 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.\cite{DS}
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\bibitem[DS]{DS} Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
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