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arXiv:2107.10298v2

Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

Dmitry Kleinbock, Anurag Rao

math.NT11J1311J8311H0637A17
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Paper metadata

Original submission
July 21, 2021
Latest revision
October 19, 2021 · version 2
Claimed MSC
11J13, 11J83, 11H06, 37A17
arXiv categories
math.NT
Paper license
CC BY 4.0 ↗
Comments
16 pages, 3 figures; the new version has a stronger version of the main theorem with a simpler proof
Journal reference
Moscow J. Comb. Number Th. 11 (2022) 97-114

Abstract

In a recent paper of Akhunzhanov and Shatskov the two-dimensional Dirichlet spectrum with respect to Euclidean norm was defined. We consider an analogous definition for arbitrary norms on R2\mathbb{R}^2 and prove that, for each such norm, the set of Dirichlet improvable pairs contains the set of badly approximable pairs, hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk--Rogers and Mahler to establish a Hajós--Minkowski type result for the critical locus of a cylinder. As a corollary, using a recent result of the first named author with Mirzadeh, we conclude that for any norm on R2\mathbb{R}^2 the top of the Dirichlet spectrum is not an isolated point.

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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.

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Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

For every norm on the plane, badly approximable pairs are correctly shown to be Dirichlet-improvable, hence the improvable set is hyperplane absolute winning; the top of the associated Dirichlet spectrum is an accumulation point. One displayed subscript is a harmless typo.

Theorem 1.1Correct

Badly approximable pairs are Dirichlet-improvable for every norm

Page 2 and Section 5 · main theorem and completion · arXiv:2107.10298v2

The Dani correspondence embeds a pair into the space of unimodular three-lattices and identifies strict improvement with eventual avoidance of the critical locus of the cylinder norm η(x1,x2,x3)=max⁡{ν(x1,x2),∣x3∣}\eta(x_1,x_2,x_3)=\max\{\nu(x_1,x_2),|x_3|\}. The Chalk--Rogers--Mahler classification places each critical lattice in a structured family. If a badly approximable orbit accumulated on that family, the displayed lattice bases would produce integer vectors violating its uniform lower bound. Thus the orbit eventually avoids the critical locus.

Corollary 1.2Correct

The top of the Dirichlet spectrum is not isolated

Page 2 and Section 5 · spectrum corollary · arXiv:2107.10298v2

The main theorem gives a hyperplane-absolute-winning set below the top value. The cited dimension-drop theorem bounds the Hausdorff dimension of vectors whose orbit avoids a fixed neighborhood of the critical locus. If the top value were isolated, all vectors in the winning set would avoid one common neighborhood, contradicting full Hausdorff dimension of a winning set.

02Proofs4 reported findingsCorrect

The dynamical reformulation, cylinder critical-locus classification, orbit-avoidance argument, and dimension-drop contradiction are correct and complete. Two local norm subscripts have uniquely determined corrections.

Sections 2--4Correct and complete

The critical-locus alternatives exhaust all cylinder lattices

Pages 3--13 · dynamical proposition and critical-locus theorem · arXiv:2107.10298v2

The critical radius normalization gives cν=rη3c_\nu=r_\eta^3. The Chalk--Rogers alternatives are translated into unimodular-lattice coordinates without losing boundary cases, and Mahler's reduction handles the exceptional planar section. Each resulting family is then checked against the diagonal action used by the Dani correspondence.

Definition of the supremum-norm improvable setTypo

The formula uses the Euclidean spectrum function

Page 2 · display defining the supremum-norm improvable set · arXiv:2107.10298v2

The set is denoted DI∞\mathbf{DI}_\infty and the surrounding paragraph discusses c∞c_\infty, but the display reads c2(x)<1c_2(\mathbf x)<1. The uniquely consistent correction is c∞(x)<1c_\infty(\mathbf x)<1; all later definitions use the norm-matched function cνc_\nu.

Proof of Theorem 1.1Typo

The limiting lattice is assigned to the wrong critical locus

Page 14 · first paragraph of the proof of Theorem 1.1 · arXiv:2107.10298v2

The proof assumes that askΛxa_{s_k}\Lambda_{\mathbf x} converges to a lattice in the critical locus of the three-dimensional cylinder norm η\eta, but the displayed membership is Λ∈Lν\Lambda\in\mathcal L_\nu. Proposition 5.1 and the ambient space X3X_3 force the correction Λ∈Lη\Lambda\in\mathcal L_\eta; the subsequent two-case argument already uses exactly that locus.

Section 5Correct and complete

Bad approximation excludes every critical-locus accumulation case

Pages 14--15 · proof of the main theorem · arXiv:2107.10298v2

In each structural alternative, convergence to the critical locus supplies primitive integer vectors whose first two coordinates and denominator violate the compact-orbit lower bound for a badly approximable pair. The estimates are uniform along the convergent subsequence, and the strict inequality in the improvability definition follows from compactness of the avoided critical locus.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2107.10298v2
Authors listed
Dmitry Kleinbock, Anurag Rao
Audit date
August 19, 2026
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