  The goal of this paper is to study the action of the group of translations
over self-similar tilings in the euclidian space $\mathbb{R}^d$. It
investigates the behaviour near zero for spectral measures for such dynamical
systems. Namely the paper gives a H\"older 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.
