The Asymptotic Distribution of Frobenius Numbers

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.