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# 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 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]

# References

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

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

11J68*no label found*

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