The asymptotic distribution of Frobenius numbers

TL;DR

This paper establishes the limiting distribution of Frobenius numbers for random integer vectors in high dimensions, linking it to lattice geometry and group dynamics, extending previous 3D results.

Contribution

It introduces a novel interpretation of Frobenius numbers via group actions and proves a new equidistribution theorem for multidimensional Farey sequences.

Findings

Frobenius number distribution converges to a limit for random vectors in expanding domains.

The limit distribution relates to the covering radius of a simplex with respect to a random lattice.

The approach combines dynamics on lattice spaces with equidistribution of Farey sequences.

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.