BADZIAHIN, D. and LEVESLEY, J.
(2007)
*A note on simultaneous and multiplicative Diophantine approximation on planar curves.*
Glasgow Mathematical Journal, 49 (2).
pp. 367-375.
ISSN 1469-509X

## Abstract

Let $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

Item Type: | Article |
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Institution: | The University of York |

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

Depositing User: | York RAE Import |

Date Deposited: | 12 Jun 2009 10:23 |

Last Modified: | 12 Jun 2009 10:23 |

Published Version: | http://dx.doi.org/10.1017/S0017089507003722 |

Status: | Published |

Publisher: | Nature Publishing Group |

Identification Number: | 10.1017/S0017089507003722 |

URI: | http://eprints.whiterose.ac.uk/id/eprint/6095 |