Discrepancy and rectifiability of almost linearly repetitive Delone sets

TL;DR

This paper extends discrepancy bounds to a broader class of Delone sets, showing that certain repetitive properties imply rectifiability and that some tilings are not almost linearly repetitive.

Contribution

It generalizes discrepancy bounds to Delone sets without finite local complexity using $ ext{epsilon}$-linear repetitivity, establishing rectifiability and non-repetitiveness of certain tilings.

Findings

Discrepancy bounds hold for $ ext{epsilon}$-linearly repetitive Delone sets.

Small $ ext{epsilon}$-linear repetitivity implies rectifiability.

Incommensurable multiscale substitution tilings are not almost linearly repetitive.

Abstract

We extend a discrepancy bound of Lagarias and Pleasants for local weight distributions on linearly repetitive Delone sets and show that a similar bound holds also for the more general case of Delone sets without finite local complexity if linear repetitivity is replaced by $\varepsilon$-linear repetitivity. As a result we establish that Delone sets that are $\varepsilon$-linear repetitive for some sufficiently small $\varepsilon$ are rectifiable, and that incommensurable multiscale substitution tilings are never almost linearly repetitive.