Quantitative weak mixing of self-affine tilings

TL;DR

This paper extends the understanding of spectral measure regularity in self-affine tiling dynamical systems, demonstrating logarithmic weak mixing rates and generalizing previous one-dimensional results to higher dimensions.

Contribution

It generalizes the log-Hölder continuity results of spectral measures from one-dimensional to higher-dimensional self-affine tilings, providing uniform spectral estimates and weak mixing rates.

Findings

Spectral measures exhibit log-Hölder continuity in higher dimensions.

Established uniform estimates across spectral parameters.

Demonstrated logarithmic weak mixing rates.

Abstract

We study the regularity of spectral measures of dynamical systems arising from a translation action on tilings of substitutive nature. The results are inspired in the work of Bufetov and Solomyak, where they established a log-H\"older modulus of continuity of one-dimensional self-similar tiling systems. We generalize this result to higher dimensions in the more general setting of self-affine tilings systems. Further analysis leads to uniform estimates in the whole space of spectral parameters, allowing to deduce logarithmic rates of weak mixing.