On the density of Birkhoff sums for Anosov diffeomorphisms

Let f be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all periodic points is dense in the whole set of real numbers.