Discrepancy and rectifiability of almost linearly repetitive Delone sets

Yotam Smilansky, Yaar Solomon

Research output: Working paper/PreprintPreprint

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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.
Original languageEnglish
StatePublished - 1 Sep 2021

Keywords

  • Mathematics - Dynamical Systems
  • Mathematics - Metric Geometry
  • 52C23
  • 37B20

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