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

Abstract

We study the number of lattice points in integer dilates of the rational polytope [formula presented], where a1,…,an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1,…,an, find the largest value of t (the Frobenius number) such that m1a1+···+mnan=t has no solution in positive integers m1,…,mn. This is equivalent to the problem of finding the largest dilate t[formula presented] such that the facet {∑k=1nxkak=t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L([formula presented],t)≔#(t[formula presented]∩[formula presented]n) and L([formula presented]°,t)≔#(t[formula presented]°∩[formula presented]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.