Discrepancy and rectifiability of almost linearly repetitive Delone sets

Yotam Smilansky, Yaar Solomon*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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
Issue number12
StatePublished - 1 Dec 2022
Externally publishedYes


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


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