Discrepancy and rectifiability of almost linearly repetitive Delone sets

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 $\epsilon$-linear repetitivity. As a result we establish that Delone sets that are $\epsilon$-linear repetitive for some sufficiently small $\epsilon$ are rectifiable, and that incommensurable multiscale substitution tilings are never almost linearly repetitive.

13 pages, 1 figure. To appear in Nonlinearity

References in corpus (3)

• A dichotomy for bounded displacement equivalence of Delone sets
• Multiscale Substitution Tilings
• Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings