# Roth’s theorem

The following theorem is due to Klaus Roth and it is a generalization^{} of a previous theorem of Liouville (see Liouville approximation theorem^{}). Roth was awarded the Fields Medal for his work on the geometry of numbers. W. M. Schmidt generalized the result even further. The result is widely used to prove that a certain number is transcendental. Here, for a rational number^{} $t$ in reduced form, the denominator of $t$ is denoted by $d(t)$.

###### Theorem 1.

For any algebraic number^{} $\alpha $ and for any $\u03f5\mathrm{>}\mathrm{0}$ there are only finitely many rational numbers $t$ with:

$$ |

In other words, the equation:

$$ |

has only finitely many solutions with $p\mathrm{\in}\mathrm{Z}$ and $q\mathrm{\in}{\mathrm{Z}}^{\mathrm{+}}$.

Title | Roth’s theorem |
---|---|

Canonical name | RothsTheorem |

Date of creation | 2013-03-22 15:02:23 |

Last modified on | 2013-03-22 15:02:23 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11J81 |

Classification | msc 11J68 |

Related topic | ExampleOfTranscendentalNumber |