The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

Abstract

We study the number of lattice points in integer dilates of the rational polytope P={(x 1,…,x n)∈ R ⩾0 n: ∑ k=1 n x ka k⩽1} , where a 1,…, a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1,…, a n , find the largest value of t (the Frobenius number) such that m 1 a 1+···+ m na n = t has no solution in positive integers m 1,…, m n . This is equivalent to the problem of finding the largest dilate t P such that the facet {∑ k=1 n x k a k = t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L( P , t)≔#( t P ∩ Z n ) and L( P °, t)≔#( t P °∩ Z n ). Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number.