Effective Limit Distribution of the Frobenius Numbers

Abstract

The Frobenius number $F(\ba)$ of a lattice point $\ba$ in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$. 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.

15 pages

References in corpus (2)

• The asymptotic distribution of Frobenius numbers
• On effective equidistribution of expanding translates of certain orbits in the space of lattices

Cited by in corpus (6)

• Translates of rational points along expanding closed horocycles on the modular surface
• Joint partial equidistribution of Farey rays in negatively curved manifolds and trees
• Effective equidistribution of primitive rational points on expanding horospheres
• Diameters of random circulant graphs
• Equidistribution of Farey sequences on horospheres in covers of SL(n+1,Z)\SL(n+1,R) and applications
• Effective equidistribution of rational points on expanding horospheres