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BiLipschitz and bounded displacement equivalence of Delone sets
Abstract
We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.
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BibTeX
@misc{smilansky2026bilipschitz,
title = {{BiLipschitz and bounded displacement equivalence of Delone sets}},
author = {Yotam Smilansky and Yaar Solomon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01763},
eprint = {2609.01763},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01763},
}
RIS
TY - UNPB
TI - BiLipschitz and bounded displacement equivalence of Delone sets
AU - Yotam Smilansky
AU - Yaar Solomon
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01763
UR - https://arxiv.org/abs/2609.01763
AB - We survey biLipschitz (BL) and bounded displacement (BD) equivalence of Delone sets, with an emphasis on examples arising in aperiodic order. After recalling the classical results of Burago-Kleiner, McMullen, and Laczkovich, we discuss criteria for rectifiability, uniform spreadness, and BD equivalence, and how these relate to discrepancy and point-counting estimates. We then focus on Delone sets associated with substitution tilings, where these questions can often be studied through the combinatorial and spectral properties of the underlying substitution rules. We also consider selected classes of cut-and-project sets and briefly discuss extensions beyond BL and BD equivalence.
ER -