Effective limit distribution of the Frobenius numbers

Abstract

The Frobenius number$F(\boldsymbol{a})$of a lattice point$\boldsymbol{a}$in$\mathbb{R}^{d}$with positive coprime coordinates, is the largest integer which cannotbe expressed as a non-negative integer linear combination of the coordinates of$\boldsymbol{a}$. Marklof in [The asymptotic distribution of Frobenius numbers, Invent. Math.181(2010), 179–207] proved the existence of the limit distribution of the Frobenius numbers, when$\boldsymbol{a}$is taken to be random in an enlarging domain in$\mathbb{R}^{d}$. We will show that if the domain has piecewise smooth boundary, the error term for the convergence of the distribution function is at most a polynomial in the enlarging factor.