Spectral measure at zero for self-similar tilings

The goal of this paper is to study the action of the group of translations over self-similar tilings in the Euclidean space $\mathbb R^d$. It investigates the behaviour near zero of spectral measures for such dynamical systems. Namely, the paper gives a Hölder asymptotic expansion near zero for these spectral measures. It is a generalization to higher dimension of a result by Bufetov and Solomyak who studied self similar-suspension flows for substitutions. The study of such asymptotics mostly involves the understanding of the deviations of some ergodic averages.