Spectral measure at zero for self-similar tilings

TL;DR

This paper analyzes the spectral measure behavior at zero for self-similar tilings in Euclidean space, providing a Hölder asymptotic expansion that generalizes previous one-dimensional results to higher dimensions.

Contribution

It introduces a higher-dimensional generalization of spectral measure asymptotics for self-similar tilings, extending prior work on substitution flows.

Findings

Hölder asymptotic expansion of spectral measures near zero

Generalization of Bufetov and Solomyak's results to higher dimensions

Insights into deviations of ergodic averages in self-similar tilings

Abstract

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.