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Paper metadata

Original submission
December 8, 2024
Claimed MSC
arXiv categories
math.NT

Abstract

Akhunzhanov and Shatskov defined the Dirichlet spectrum, corresponding to m×nm \times n matrices and to norms on Rm\mathbb{R}^m and Rn\mathbb{R}^n. In case (m,n)=(2,1)(m,n) = (2,1) and using the Euclidean norm on R2\mathbb{R}^2, they showed that the spectrum is an interval. We generalize this result to arbitrary (m,n)≠(1,1)(m,n) \neq (1,1) and arbitrary norms, improving previous works from recent years. We also define some related spectra and show that they too are intervals. Our argument is a modification of an argument of Khintchine from 1926.

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Audited against arXiv v1

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
01Statements4 reported findingsCorrect

The interval theorem for Dirichlet spectra in every matrix dimension other than (1,1)(1,1), its weighted lattice-flow extension, and the stated ψ\psi-spectrum consequences are correct. The printed upper-uniformity construction fails at rational line parameters, but a direct uniform Dirichlet estimate repairs both affected applications. The remaining notation errors are mechanically identifiable as typos.

Theorem 1Correct

The full Dirichlet spectrum is an interval outside dimension (1,1)(1,1)

Page 2 and pages 6–14 · Theorem 1 and its deduction from Theorems 2 and 5 · arXiv:2412.05858v1

For the compacta Kε={Λ:λ1(Λ)≥ε}K_\varepsilon=\{\Lambda:\lambda_1(\Lambda)\ge\varepsilon\}, the extended threshold equals the limsup of the first minimum along the diagonal orbit. Lemma 2 identifies its appropriate power with the limsup defining the Dirichlet constant. The topological theorem produces every interior value from the dense full-measure set of maximal limsup points and the dense connected family on which the limsup is zero. The zero endpoint is realized on those rational affine families, while the maximal endpoint is realized on a full-measure set. This gives Dm,n=[0,Δ]\mathbb D_{m,n}=[0,\Delta] and uncountably many realizations in every open set when max⁡(m,n)>1\max(m,n)>1.

Theorem 2Correct

Weighted extended spectra for arbitrary compact exhaustions

Page 3 and pages 10–14 · Theorem 2 and Section 5.1 · arXiv:2412.05858v1

Composing the continuous exhaustion gauge with gt,α,βΛΘg_{t,\boldsymbol\alpha,\boldsymbol\beta}\Lambda_\Theta gives the continuous function required by Theorem 5. For n=1n=1, rational affine lines supply the upper-uniform family. The direct Dirichlet estimate recorded under Proofs is uniform in the line parameter and tends to zero because αmax⁡<1\alpha_{\max}<1. For n≥2n\ge2, rational affine planes Θi=z\Theta\mathbf i=\mathbf z contain a fixed contracting lattice vector. Kleinbock–Weiss equidistribution shows that almost every expanding-horosphere point visits every EcE_c with c<bc<b at arbitrarily large times. The endpoint arguments then give the entire interval [0,b][0,b].

Theorem 3 and Corollary 1Correct

The ψ\psi-Dirichlet spectrum and prescribed uniform exponent

Pages 4–5 and 14–20 · Theorem 3, Corollary 1, and technical lemmas · arXiv:2412.05858v1

The same connected-family argument applies to λΘ,ψ\lambda_{\Theta,\psi}. When n=1n=1, the additional condition tmψ(t)→∞t^m\psi(t)\to\infty is exactly ψ(t)−1/m=o(t)\psi(t)^{-1/m}=o(t); the repaired direct Dirichlet estimate under Proofs then tends uniformly to zero. For n≥2n\ge2, the fixed rational kernel vector contracts as t−1/nt^{-1/n}. Lemma 1 transfers the almost-everywhere positive limsup for the Dirichlet scale t−1t^{-1} to infinite limsup at the smaller function ψ=o(t−1)\psi=o(t^{-1}), supplying the dense maximal set. Lemma 2 then converts the power-law choice of ψ\psi into the claimed simultaneous prescription of the uniform exponent and the critical limsup.

Definitions of $\chi$ and $\chi_\gamma$Typo

The approximating vector is printed in the wrong ambient set

Pages 1 and 4 · definitions of χ(Θ,t)\chi(\Theta,t) and χγ(Θ,t)\chi_\gamma(\Theta,t) · arXiv:2412.05858v1

Both displays place p\mathbf p in Rm\mathbb R^m. Literally, the minimum would then be zero by choosing p=Θq\mathbf p=\Theta\mathbf q. Every surrounding statement, formula (4), the definition of λΘ,ψ\lambda_{\Theta,\psi}, Lemma 2, and every proof instead use p∈Zm\mathbf p\in\mathbb Z^m. Replacing Rm\mathbb R^m by Zm\mathbb Z^m in the two displays is therefore a uniquely determined symbol correction. Under the audit’s typo rule, it does not make the central theorems incorrect.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The topological interpolation theorem and the technical comparison lemmas are correct. In both n=1n=1 homogeneous-space applications, however, the printed proof asks for an infinite strictly improving approximation sequence even when the line parameter is rational, where no such sequence exists. A direct Dirichlet estimate gives a verified uniform repair. The remaining notation and indexing slips are local and harmless.

Theorem 5Correct and complete

Nested connected interpolation realizes every intermediate limsup

Pages 8–10 · Theorem 5 and its proof · arXiv:2412.05858v1

At each stage a point with value above cc is joined, inside one connected member of the distinguished family, to a point whose entire late-time tail is below c−εkc-\varepsilon_k. The two closed subsets defined by staying below c−εk+1c-\varepsilon_{k+1} and reaching at least cc cannot separate that connected set, so an intermediate point and a smaller neighborhood satisfy both the upper bound and a near-cc witness. Compactly nested closures give a common point; the covering time intervals bound its eventual values by cc, and the witnesses force limsup cc. Diagonal avoidance of a purported enumeration proves uncountability.

The $n=1$ upper-uniformity estimatesIncorrect as written · verified repair

The strictly improving sequence does not exist for rational parameters, but a direct estimate repairs both uses

Pages 11–15 · n=1n=1 cases in the proofs of Theorems 2 and 3 · arXiv:2412.05858v1

Both applications choose an infinite sequence qkq_k with strictly decreasing errors ∣qky−pk∣|q_ky-p_k|. When y∈Qy\in\mathbb Q, an exact zero error occurs and no infinite strictly decreasing sequence exists, so the subsequent balancing times are undefined; rational values of yy do occur on the chosen affine lines. For any integer N≥1N\geq1, Dirichlet's theorem instead gives 1≤q≤N1\leq q\leq N and ∣qy−p∣≤1/N|qy-p|\leq1/N, uniformly in yy. In the proof of Theorem 2, take NN comparable to t(1+αmax⁡)/2t^{(1+\alpha_{\max})/2}. Then the constructed lattice vector satisfies max⁡{qt, imax⁡tαmax⁡∣qy−p∣}≤Ct(αmax⁡−1)/2⟶0.\max\left\{\frac qt,\ i_{\max}t^{\alpha_{\max}}|qy-p|\right\}\leq C t^{(\alpha_{\max}-1)/2}\longrightarrow0. In the proof of Theorem 3, take NN comparable to (tψ(t)−1/m)1/2\bigl(t\psi(t)^{-1/m}\bigr)^{1/2} to obtain max⁡{qt, imax⁡ψ(t)−1/m∣qy−p∣}≤C(ψ(t)−1/mt)1/2⟶0.\max\left\{\frac qt,\ i_{\max}\psi(t)^{-1/m}|qy-p|\right\}\leq C\left(\frac{\psi(t)^{-1/m}}t\right)^{1/2}\longrightarrow0. The constants are fixed on each rational affine line, so both estimates are uniform on compact subsets and verify the required upper uniformity for rational and irrational yy at once. The two auxiliary claims should correspondingly require their threshold to be uniform on compact subsets; the repaired estimates satisfy that corrected premise. For n≥2n\geq2, the printed fixed-kernel-vector argument is unaffected, and the equidistribution step supplies the dense maximal-limsup set.

Lemmas 1–4Correct and complete

Successive minimizers and limsup comparison

Pages 16–20 · technical lemmas and proofs · arXiv:2412.05858v1

Successive minimizing pairs have strictly increasing denominator norms and strictly decreasing errors. Their balancing times are precisely the local maxima of λΘ,ψ\lambda_{\Theta,\psi}. If λΘ,ψ\lambda_{\Theta,\psi} stayed bounded while ψ=o(t−1)\psi=o(t^{-1}), the products ∥qk+1∥n∥Θqk−pk∥m\lVert\mathbf q_{k+1}\rVert^n\lVert\Theta\mathbf q_k-\mathbf p_k\rVert^m would tend to zero; balancing the same pairs at the Dirichlet scale would then force lim sup⁡λΘ,t−1=0\limsup\lambda_{\Theta,t^{-1}}=0, a contradiction. The analogous balancing calculation for ψγ\psi_\gamma yields exactly the exponent 1+γn/m1+\gamma n/m and proves the comparison with χγn/m\chi_{\gamma n/m}.

Section 5.2 line familyTypo

One occurrence of Zm\mathbb Z^m must be Qm\mathbb Q^m

Page 14 · first paragraph of the n=1n=1 case in the proof of Theorem 3 · arXiv:2412.05858v1

The proof says “as before” but prints the line family with both i,z∈Zm\mathbf i,\mathbf z\in\mathbb Z^m; the earlier locally connected family has i∈Zm\mathbf i\in\mathbb Z^m and z∈Qm\mathbf z\in\mathbb Q^m. The subsequent sentence invokes the earlier local-connectedness proof and the construction again chooses a denominator clearing z\mathbf z, so Qm\mathbb Q^m is the intended set. Restoring it makes the cited argument apply verbatim.

Proof of Theorem 5Minor formal correction

Two compact-set and enumeration indices are locally misstated

Pages 8–10 · base step and final uncountability paragraph · arXiv:2412.05858v1

Upper uniformity applies on compact sets, so the base-step bound should be stated on Xj0∩Ω0‾X_{j_0}\cap\overline{\Omega_0} rather than all of Xj0X_{j_0}; this is exactly the set used later. In the diagonal argument, exclude vkv_k when choosing the kkth smaller neighborhood rather than the printed vk+1v_{k+1}. Both corrections are internal to the proof, mechanically preserve its induction, and change no theorem statement.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2412.05858v1
Authors listed
Alon Agin, Barak Weiss
Audit date
August 19, 2026
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