Effective Limit Distribution of the Frobenius Numbers

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.