Discrepancy and rectifiability of almost linearly repetitive Delone sets

Yotam Smilansky, Yaar Solomon

Research output: Contribution to journalArticlepeer-review

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

Original languageEnglish
Pages (from-to)6204-6217
Number of pages14
JournalNonlinearity
Volume35
Issue number12
DOIs
StatePublished - Oct 2022

Keywords

  • 52C23, 37B20
  • aperiodic order
  • biLipschitz equivalence
  • Delone sets
  • linear repetitivity
  • mathematical quasicrystals

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy (all)
  • Applied Mathematics

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