The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums, Original Research Article

We study the number of lattice points in integer dilates of the rational polytope image, 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 timage such that the facet {∑k=1nxkak=t} contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials L(image,t)colon, equals#(timage∩imagen) and L(image°,t)colon, equals#(timage°∩imagen). 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 Zagierʹs higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier–Dedekind sums and use them to give new bounds for the Frobenius number.