Asymptotics of cone theta functions, Gauss sums over parallelepipeds, and generalized Gram relations for polyhedra

Abstract: 

We begin with cone theta functions, which are cousins of theta functions, but are defined as a natural discretization of the volume of a spherical polytope. We obtain precise asymptotics of the cone theta function attached to any simplicial cone, near a rational "cusp", and use these asymptotics to give new extensions of the Gram-relations for the solid angles of faces of a simple rational polytope.