On the limit distribution of Frobenius numbers

TL;DR

This paper investigates the asymptotic tail behavior and distribution concentration of Frobenius numbers for random integer vectors, using geometric number theory techniques to establish limit distribution properties.

Contribution

It provides the first asymptotic formula for the tail behavior of the Frobenius number's limit distribution and shows uniform bounds and concentration phenomena in high dimensions.

Findings

Asymptotic tail behavior formula for the limit distribution.

Uniform upper bounds on the probability of large Frobenius numbers.

Concentration of the distribution's mass in a short interval for large dimensions.

Abstract

The Frobenius number g(a) of an integer vector a with positive coprime coefficients is defined as the largest integer that does not have a representation as a non-negative integer linear combination of the coefficients of a. According to a recent result by Marklof, if a is taken to be random in an expanding d-dimensional domain D, then g(a) (appropriately rescaled) has a limit distribution. In the present paper we prove an asymptotic formula for the (algebraic) tail behavior of this limit distribution. We also prove that the corresponding upper bound on the probability of the Frobenius number being large holds uniformly with respect to the expansion factor of the domain D. Finally we prove that for large d, the limit distribution has almost all of its mass concentrated in a fairly short interval. The techniques involved in the proofs come from the geometry of numbers, and in particular we use results by Schmidt on the distribution of sublattices of Z^m, and bounds by Rogers and Schmidt on lattice coverings of space with convex bodies.