The asymptotic distribution of Frobenius numbers

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.

19 pages

Cited by in corpus (10)

• Effective Limit Distribution of the Frobenius Numbers
• Integer points and their orthogonal lattices
• Equidistribution of primitive rational points on expanding horospheres
• Distribution of Shapes of orthogonal Lattices
• Integer points on spheres and their orthogonal grids
• Skinning measures in negative curvature and equidistribution of equidistant submanifolds
• Translates of rational points along expanding closed horocycles on the modular surface
• Joint partial equidistribution of Farey rays in negatively curved manifolds and trees
• Extreme events for horocycle flows
• Matrix Kloosterman sums