Exact Regularity and the Cohomology of Tiling Spaces

TL;DR

This paper extends the concept of exact regularity from homological Pisot substitutions to general tiling spaces, providing uniform frequency estimates and convergence rates, with specific results for substitution and one-dimensional tilings.

Contribution

It generalizes the Exact Regularity Property to arbitrary tiling spaces and quantifies convergence rates and measure constraints, advancing understanding of tiling space cohomology.

Findings

Existence of k patches governing all patch appearances.

Uniform estimates on frequency convergence.

Quantitative bounds for substitution and one-dimensional tilings.

Abstract

The Exact Regularity Property was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider exact regularity for arbitrary tiling spaces. Let ${T}$ be a $d$ dimensional repetitive tiling, and let $\Omega_{{T}}$ be its hull. If $\check H^d(\Omega_{{T}}, Q) = Q^k$, then there exist $k$ patches whose appearance govern the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling ${T}$ comes from a substitution, then we can quantify that convergence rate. If ${T}$ is also one-dimensional, we put constraints on the measure of any cylinder set in $\Omega_{{T}}$.