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 |