The interplay between dynamical systems and number theory

Those number theoretical results or problems are quite simple to state and have one point in common. It turns out that they can be reinterpreted and investigated with methods from discrete dynamical systems and ergodic theory: Mahler measures are topological entropies, Hindman's theorem is a consequence of a topological modification of ideas related to Poincaré recurrence, Margulis discussed invariant measures of group actions on Lie groups and the work of Green and Tao combines ergodic Ramsey theory in the spirit of Furstenberg's approach to Szeméredi's theorem with deep results from analytic number theory.