Prescribed absolute values, character sums and spectrum integrality

Let G be a group and let ρreg be the complex left-regular representation of G. We consider the following problem: For which inverse-closed subsets S⊆G the spectrum of the matrix ∑g∈Sρreg(g) is integral? (that is, all of the eigenvalues of the matrix are integers). For abelian G, a complete characterization, due to Bridges and Mena, is known. Here we are interested in the case where G is the dihedral group, D2n, which turns out to be a far-reaching generalization of the abelian case. We obtain a complete characterization when n is an odd prime and present partial results when n is a prime power.