The arithmetic of rational polytopes

We study the number of integer points (”lattice points”) in rational polytopes. We use an associated generating function in several variables, whose coefficients are the lattice point enumerators of the dilates of a polytope. We focus on applications of this theory to several problems in combinatorial number theory. In chapter 2, we present a new method of deriving the lattice point count operators for rational polytopes. In particular, we show how various generalizations of Dedekind sums appear naturally in the lattice point count formulas, and give geometric interpretations of reciprocity laws for these sums. In chapter 3, we use our methods to obtain new results on the Frobenius problem: namely, given positive integers a1,..., an with gcd(a1,..., an) = 1, find the largest integer that cannot be represented as a linear combination of a1,..., an with nonegative coefficients. We transfer this problem into our geometric setting and deduce and extend from this point of view some classical results on this problem. In our formulas, the following generalization of the Dedekind sum appears naturally: Let c1,..., cn ∈ Z be relatively prime to c ∈ Z, and let t ∈ Z. Define the Fourier-Dedekind sum as σt (c1,..., cn; c)