We consider self-similar tilings of the plane, such as the Penrose tiling, and the associated dynamical systems. These systems are uniquely ergodic, which means that they have a unique invariant probability measure. Roughly speaking, this corresponds to the fact that every tile appears in the tiling with a well-defined uniform frequency. It is of interest to study how quickly the averages converge to this frequency, whether one can interpret the deviation from the mean in some way and whether it satisfies some natural limit law. In a joint work with A. Bufetov, we perform such an analysis for a large class of self-similar tilings.
Questions of this kind arose in relation to a problem posed by D. Burago and B. Kleiner in 1998: is the set of vertices of the Penrose tiling (or more general self-similar tilings) bi-Lipschitz equivalent to the lattice? Positive answer was recently given by Y. Solomon, and I will discuss this connection briefly.