Research output: Contribution to journal › Article

Journal | Journal of Number Theory |
---|---|

Date | Accepted/In press - 29 Oct 2015 |

Date | E-pub ahead of print - 8 Dec 2015 |

Date | Published (current) - 1 May 2016 |

Volume | 162 |

Number of pages | 12 |

Pages (from-to) | 11-22 |

Early online date | 8/12/15 |

Original language | English |

We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R^{2}: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q_{0}, there exists p/q∈Qd such that 1≤q≤Q and x-p/q≤κq^{-a}Q^{-A}. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q_{0}=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.

© 2015 Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.

- Diophantine approximation, Dirichlet's theorem

## Programme Grant-New Frameworks in metric Number Theory

Project: Research project (funded) › Research

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