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Journal | Transactions of the American Mathematical Society |
---|---|

Date | Accepted/In press - 5 May 2016 |

Date | E-pub ahead of print - 19 Jul 2017 |

Date | Published (current) - Jan 2018 |

Volume | 370 |

Number of pages | 23 |

Pages (from-to) | 577-599 |

Early online date | 19/07/17 |

Original language | English |

We investigate the question of how well points on a nondegenerate $k$-dimensional submanifold $M \subseteq \mathbb R^d$ can be approximated by rationals also lying on $M$, establishing an upper bound on the "intrinsic Dirichlet exponent" for $M$. We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of $k$ and $d$ which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on $M$ distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the Simplex Lemma for rationals lying on $M$ which provides new insights on the local distribution of rational points on nondegenerate manifolds.

© 2017 American Mathematical Society. This paper is split off from arXiv:1405.7650v2

- math.NT

## Programme Grant-New Frameworks in metric Number Theory

Project: Research project (funded) › Research

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