Research output: Contribution to journal › Article › peer-review

Journal | Annals of Mathematics |
---|---|

Date | E-pub ahead of print - 2 Apr 2009 |

Date | Published (current) - Jan 2012 |

Issue number | 1 |

Volume | 175 |

Number of pages | 49 |

Pages (from-to) | 187-235 |

Early online date | 2/04/09 |

Original language | English |

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of $R^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds -- a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near non-degenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).

- Number Theory, simultaneous Diophantine approximation on manifolds, , metric theory, Khintchine theorem, Jarnik theorem, Hausdorff dimension, ubiquitous systems, rational points near manifolds

## Geometrical, dynamical and transference principles in non-linear Diophantine approximation and applications

Project: Research project (funded) › Research

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