Spectra of metric graphs and crystalline measures

The geometric optics trace formula gives the singular support of wave trace on a compact Riemannian manifold. In the case of of a one dimensional singular manifold, that is a metric (or quantum) graph, this formula is exact and yields a crystalline measure generalizing the Poisson and related Summation Formulae. We examine the additive structure of the spectra of such metric graphs.The resulting measures are exotic and resolve a number of problems about crystalline measures.A key ingredient in the analysis is the diophantine theory of a torus (uniform versions of Conjectures of Lang and generalizations). Joint work with P. Kurasov.