# 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

$$ |

where

$$ |

${C}_{0}$ is any fixed number greater than $\frac{160}{9}$ and $H(\alpha )$ is the of $\alpha $.[DS]

The * of the rational or quadratic irrational number* $\alpha $ is

$$H(\alpha )=\mathrm{max}(|x|,|y|,|z|)$$ |

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.

Title | Davenport-Schmidt theorem |
---|---|

Canonical name | DavenportSchmidtTheorem |

Date of creation | 2013-03-22 13:32:58 |

Last modified on | 2013-03-22 13:32:58 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 9 |

Author | Daume (40) |

Entry type | Theorem |

Classification | msc 11J68 |