Beresnevich, V., Dickinson, D., Velani, S.L. and Vaughan, R.C.
(2007)
*Diophantine approximation on planar curves and the distribution of rational points (with an appendix "Sum of two squares near perfect squares" by R.C. Vaughan).*
Annals of Mathematics, 166 (2).
pp. 367-426.
ISSN 0003-486X

## Abstract

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Item Type: | Article |
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Academic Units: | The University of York > Mathematics (York) |

Depositing User: | York RAE Import |

Date Deposited: | 18 Sep 2009 09:34 |

Last Modified: | 18 Sep 2009 09:34 |

Status: | Published |

Publisher: | Princeton University |

URI: | http://eprints.whiterose.ac.uk/id/eprint/5584 |

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