Habegger, Philipp.
(2017)
* Diophantine Approximations on Definable Sets.*
Selecta mathematica, 24.
pp. 1633-1675.

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Official URL: https://edoc.unibas.ch/59217/

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## Abstract

Consider the vanishing locus of a real analytic function on $\mathbb{R}^n$ restricted to $[0,1]^n$. We bound the number of rational points of bounded height that approximate this set very well. Our result is formulated and proved in the context of o-minimal structure which give a general framework to work with sets mentioned above. It complements the theorem of Pila-Wilkie that yields a bound of the same quality for the number of rational points of bounded height that lie on a definable set. We focus our attention on polynomially bounded o-minimal structures, allow algebraic points of bounded degree, and provide an estimate that is uniform over some families of definable sets. We apply these results to study fixed length sums of roots of unity that are small in modulus.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Zahlentheorie (Habegger) |
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UniBasel Contributors: | Habegger, Philipp |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Springer |

ISSN: | 1022-1824 |

e-ISSN: | 1420-9020 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Identification Number: | |

Last Modified: | 24 Jul 2020 13:44 |

Deposited On: | 24 Jul 2020 13:44 |

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