Dirichlet series and simultaneous observability : two problems solved by the same approach

In a classical paper, Ingham gave a simple proof of an important theorem of Pólya on singular points of Dirichlet series under a uniform gap assumption on the exponents. Bernstein generalized Pólya's theorem by weakening this gap condition. We give a simpler proof of Bernstein's theorem by applying a recent generalization of Ingham's theorem. Furthermore, we also solve a simultaneous observability problem using this theory.