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 k a 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 n a 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.