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Davenport-Schmidt theorem
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(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
where
$C_0$ 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 $$H(\alpha)=\operatorname{max}(|x|,|y|,|z|)$$ 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]
- DS
- Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. Acta Arithmetica XIII, 1967.
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"Davenport-Schmidt theorem" is owned by Daume.
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Cross-references: coefficients, integral, relatively prime, equation, number, 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 2070 times total.
Classification:
| AMS MSC: | 11J68 (Number theory :: Diophantine approximation, transcendental number theory :: Approximation to algebraic numbers) |
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Pending Errata and Addenda
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