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Paper metadata

Original submission
October 20, 2026
Claimed MSC
52C23, 51F30, 11J71
arXiv categories
math.MG, math.DS, math.NT
Paper license
CC BY 4.0 ↗
Comments
26 pages

Abstract

We construct a cut-and-project set in a totally irrational plane in R3\mathbb{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.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements2 reported findingsNo statement issues found

The construction of a regular cut-and-project Delone set with countable window boundary that is not bi-Lipschitz equivalent to a lattice, and its uniform-density consequence, are supported.

Theorem 1No issue found

The constructed regular model set is non-rectifiable

PDF pp. 2–3 and 6–25 · arXiv:2610.27640v1

The nested interval construction makes 1,α1,α21,\alpha_1,\alpha_2 rationally independent and produces a compact window that is the closure of its interior with countable boundary. The final boundary translation avoids every lattice internal coordinate. At each prescribed bi-Lipschitz scale, the associated strip pattern realizes the finite stretched-pair obstruction, so no bi-Lipschitz equivalence to any lattice can exist.

Corollary 2No issue found

The example nevertheless has uniform asymptotic density

PDF pp. 4 and 17–19 · arXiv:2610.27640v1

The row-counting lemma is uniform in the square's position and compares the number of selected lattice points with the window length. After the linear identification with the physical plane, the density is ∣W∣/∣nα∣|W|/|n_\alpha|. Hence every encoded limiting density is constant and therefore realizable.

02Proofs2 reported findingsNo proof issues found

The reduction to a subset of Z2\mathbb Z^2, finite-scale stretched-pair obstruction, uniform counting estimates, nested strip construction, and final boundary avoidance are correct and complete.

Proposition 7No issue found in the reviewed proof

The finite configurations obstruct every prescribed bi-Lipschitz constant

PDF pp. 8–16 · Sections 3–4 · arXiv:2610.27640v1

The stretched-pair lemma converts the alternating adjacent-square count contrast into a quantitative separation distortion. Compactness of normalized Delone configurations turns the planar statement into a finite-scale obstruction with constants uniform over the required covering-radius class.

Theorem 3 and final constructionNo issue found in the reviewed proof

The stages do not interfere and the limiting window has the claimed topology

PDF pp. 17–25 · Sections 5–6 · arXiv:2610.27640v1

Previously built patterns are controlled by the uniform row estimate, while later additions occupy intervals of total length smaller than the active contrast scale. The nested choices converge to a totally irrational slope; the endpoints accumulate only at the designated countable set, and the final translation simultaneously avoids all lattice internal coordinates.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.27640v1
Authors listed
Alan Haynes
Audit date
October 23, 2026
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