## Abstract

The following Khintchine-type theorem is proved for manifolds M embedded in R^{ k} which satisfy some mild curvature conditions. The inequality |q·x| <?(|q|) where ?(r) ? 0 as r ? 8 has finitely or infinitely many solutions qeZ^{ k} for almost all (in induced measure) points x on M according as the sum S_{ r}^{ = 1/8} ?(r)r^{ k-2} converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point in M satisfying the inequality infinitely often when ?(r) =r^{ -t} . t >k - 1. © 1990 Indian Academy of Sciences.

Original language | English |
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Pages (from-to) | 221-229 |

Number of pages | 9 |

Journal | Proceedings of the Indian Academy of Sciences - Mathematical Sciences |

Volume | 100 |

Issue number | 3 |

DOIs | |

Publication status | Published - Dec 1990 |

## Keywords

- Hausdorff dimension
- Khintchine's theorem
- manifolds
- Metric diophantine approximation