Effective Limit Distribution of the Frobenius Numbers

TL;DR

This paper investigates the distribution of Frobenius numbers for random lattice points, establishing a polynomial error bound for the convergence of their limit distribution when points are chosen from expanding domains.

Contribution

It proves that the convergence of the Frobenius number distribution has a polynomial error term for domains with piecewise smooth boundaries, extending previous results.

Findings

Error term for distribution convergence is polynomial in the enlarging factor

Limit distribution of Frobenius numbers exists for random lattice points

Results apply to domains with piecewise smooth boundary

Abstract

The Frobenius number $F(\ba)$ of a lattice point $\ba$ in $\R^d$ with positive coprime coordinates, is the largest integer which can $not$ be expressed as a non-negative integer linear combination of the coordinates of $\ba$. Marklof in \cite{M} proved the existence of the limit distribution of the Frobenius numbers, when $\ba$ is taken to be random in an enlarging domain in $\R^d$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution functions is at most a polynomial in the enlarging factor.