A positive proportion Livshits theorem

Given a transitive Anosov diffeomorphism or flow on a closed connected Riemannian manifold $M$, the Livshits theorem states that a Hölder function $\phi :M\to \mathbb{R}$ is a coboundary if all of its periods vanish. We explain how a finer statistical understanding of the distribution of these periods can be used to obtain a stronger version of the classical Livshits theorem where one only has to check that the periods of $\phi$ vanish on a set of positive asymptotic upper density. We also include a strengthening of the nonpositive Livshits theorem.