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PreprintCC BY 4.0arXiv

A cut-and-project set which is not bi-Lipschitz equivalent to a lattice

  • Alan Haynes
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Abstract

We construct a cut-and-project set in a totally irrational plane in ℝ3, with a compact window which is the closure of its interior and has countable boundary, which is not bi-Lipschitz equivalent to a lattice. This answers a question of Burago and Kleiner about cut-and-project sets, and shows in particular that the hypothesis of boundary measure zero is not enough to guarantee bi-Lipschitz equivalence to a lattice. Since our set has uniform density, it also shows that a non-rectifiable Delone set need not encode a non-realisable density, answering a question of Bhat and Dymond.

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Cite

BibTeXRIS
BibTeX
@misc{haynes2026cutandproject,
  title = {{A cut-and-project set which is not bi-Lipschitz equivalent to a lattice}},
  author = {Alan Haynes},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27640},
  eprint = {2610.27640},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27640},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A cut-and-project set which is not bi-Lipschitz equivalent to a lattice
AU  - Alan Haynes
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27640
UR  - https://arxiv.org/abs/2610.27640
AB  - We construct a cut-and-project set in a totally irrational plane in ℝ^3, with a compact window which is the closure of its interior and has countable boundary, which is not bi-Lipschitz equivalent to a lattice. This answers a question of Burago and Kleiner about cut-and-project sets, and shows in particular that the hypothesis of boundary measure zero is not enough to guarantee bi-Lipschitz equivalence to a lattice. Since our set has uniform density, it also shows that a non-rectifiable Delone set need not encode a non-realisable density, answering a question of Bhat and Dymond.
ER  -