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Paper metadata

Original submission
March 30, 2020
Latest revision
January 12, 2021 · version 2
Claimed MSC
arXiv categories
math.MG, math.NT
Comments
new section added

Abstract

Let KK be a bounded convex domain in R2\mathbb{R}^2 symmetric about the origin. The critical locus of KK is defined to be the (non-empty compact) set of lattices ΛΛ in R2\mathbb{R}^2 of smallest possible covolume such that Λ∩K={0}Λ\cap K= \lbrace 0\rbrace. These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between 00 and 11.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections 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 August 19, 2026
01Statements2 reported findingsCorrect

Every nonempty closed subset of the critical circle of a strictly convex irreducible planar domain is correctly realized as the full critical locus of a larger symmetric convex domain. The disc and arbitrary-dimension consequences follow.

Main realization theoremCorrect

Closed subsets of an irreducible critical circle are realizable

Pages 3 and 7--8 · main theorem and its proof · arXiv:2003.13829v2

The critical lattices of the irreducible domain are continuously parameterized by one boundary contact point and the next contact point. For each complementary interval of the prescribed closed parameter set, adjoining the tangent-bounded curvilinear region moves exactly that open boundary arc into the interior. Lattices with parameters in the closed set retain the three boundary contacts and remain admissible; every excluded parameter acquires an interior lattice point. The critical determinant comparison then identifies the critical locus exactly.

Disc corollaryCorrect

Arbitrary closed subsets and Hausdorff dimensions occur

Pages 2--3 · unit-disc specialization · arXiv:2003.13829v2

The unit disc is strictly convex and irreducible, and its critical locus is the rotation circle of the hexagonal lattice. Applying the realization theorem to any nonempty closed subset of that circle gives the stated domain. Choosing a closed subset of any prescribed Hausdorff dimension in [0,1][0,1] yields every dimension between 00 and 11.

02Proofs3 reported findingsCorrect

The critical-locus parametrization, tangent-region construction, and determinant comparison are correct. A repeated inequality in the comparison lemma has a unique typographical correction.

Critical-circle parametrizationCorrect and complete

Mahler's contact-point lemmas give a genuine homeomorphism

Pages 5--6 · proof of the critical-locus parametrization theorem · arXiv:2003.13829v2

Uniqueness of the critical lattice through each boundary point makes the parametrization well-defined. The interlacing lemma forces continuity of the next contact point, and compactness plus injectivity turns the descended circle map into a homeomorphism. The parallelogram exception is excluded exactly where the six-contact argument would fail.

Comparison lemmaTypo

The first critical-determinant inequality repeats the wrong symbol

Page 6 · proof of Lemma 3.1 · arXiv:2003.13829v2

The proof prints Δ(H)≤Δ(H)\Delta(H)\leq\Delta(H) after assuming H⊂KH\subset K. Admissibility gives Δ(H)≤Δ(K)\Delta(H)\leq\Delta(K). The next sentence uses a lattice that is both HH-critical and KK-admissible to prove the reverse inequality, so replacing the second HH by KK is uniquely determined and restores the displayed equality used below.

Main constructionCorrect and complete

The admissible-lattice characterization is applied in both directions

Pages 7--8 · proof of the main realization theorem · arXiv:2003.13829v2

At retained parameters, the three vectors p(t)\mathbf p(t), q(t)\mathbf q(t), and q(t)−p(t)\mathbf q(t)-\mathbf p(t) remain on the new boundary, so the contact-point criterion gives admissibility. At an omitted parameter, p(t)\mathbf p(t) lies in the added interior region, so the corresponding lattice is not admissible. Since one retained lattice is admissible, the corrected comparison lemma gives equality of critical determinants and completes the exact-locus argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2003.13829v2
Authors listed
Dmitry Kleinbock, Anurag Rao, Srinivasan Sathiamurthy
Audit date
August 19, 2026
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