### Abstract

The Frobenius number F(a) of an integer vector a with positive coprime coefficients is defined as the largest number that does not have a representation as a positive integer linear combination of the coefficients of a. We show that if a is taken to be random in an expanding d-dimensional domain, then F(a) has a limit distribution, which is given by the probability distribution for the covering radius of a certain simplex with respect to a (d−1)-dimensional random lattice. This result extends recent studies for d=3 by Arnold, Bourgain-Sinai and Shur-Sinai-Ustinov. The key features of our approach are (a) a novel interpretation of the Frobenius number in terms of the dynamics of a certain group action on the space of d-dimensional lattices, and (b) an equidistribution theorem for a multidimensional Farey sequence on closed horospheres.

Translated title of the contribution | The asymptotic distribution of Frobenius numbers |
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Original language | English |

Pages (from-to) | 179 - 207 |

Number of pages | 29 |

Journal | Inventiones Mathematicae |

Volume | 181 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2010 |

### Bibliographical note

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## Cite this

Marklof, J. (2010). The asymptotic distribution of Frobenius numbers.

*Inventiones Mathematicae*,*181*(1), 179 - 207. https://doi.org/10.1007/s00222-010-0245-z