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'Davenport-Schmidt theorem'
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| Title of object: |
Davenport-Schmidt theorem |
| Canonical Name: |
DavenportSchmidt |
| Type: |
Theorem |
| Created on: |
2003-04-04 20:04:23 |
| Modified on: |
2004-02-07 22:47:17 |
| Classification: |
msc:11J68 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
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}
\mid \xi - \alpha \mid < C\cdot H(\alpha)^{-3},
\end{displaymath}
where
\begin{displaymath}
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.
\end{displaymath}
$C_0$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the \PMlinkescapetext{height} of $\alpha$.
\begin{thebibliography}{1}
\bibitem{DS} Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
\end{thebibliography} |
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