arXiv:2606.05096v1

Khintchine's Theorem for Symmetric matrices via Flows on the Space of Symplectic Lattices

Minchang Kim

math.NTmath.DS11J8337A1711H06

Paper metadata

Original submission: June 3, 2026
Claimed MSC: 11J83, 37A17, 11H06
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 19 pages

Abstract

We establish Diophantine approximation results for real symmetric matrices by collections of linearly independent integer vectors. For $X \in \mathrm{Sym}_d(\mathbb{R})$ , we prove a Dirichlet-type theorem guaranteeing the existence of integral Lagrangian frames $(Q, P) \in \mathrm{Mat}_{d \times 2d}(\mathbb{Z})$ that satisfy $\lVert QX + P \rVert_{\mathrm{op}} \leq c_d/N$ and $\lVert Q \rVert_{\mathrm{op}} \leq N$ for any $N \geq 1$ . Furthermore, we establish a Khintchine-type zero-one law, demonstrating that the size of the set of $ψ$ -approximable symmetric matrices is determined by the convergence or divergence of the series $\sum_{q \geq 1} q^{ς- 1}ψ(q)^ς$ , where $ς= d(d+1)/2$ . The proofs rely on the reduction theory of the Siegel upper half-space, dynamical formulation over the space of symplectic lattices, and an analysis of the Siegel transform adapted to count Lagrangian frames instead of single lattice points.

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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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements2 reported findingsContains unsupported statements

The Dirichlet theorem for integral Lagrangian frames is supported. The Khintchine zero-one law is not verified because the final dynamical argument applies an ambient Haar-almost-everywhere theorem directly to a fixed unstable-horospherical parameter set without the required transfer theorem.

Theorem 1.1Correct

Dirichlet approximation by Lagrangian frames

Pages 1–2 and 7–10 · reduction-theoretic proof · arXiv:2606.05096v1

Siegel reduction produces a symplectic lattice representative with a short primitive Lagrangian frame. Undoing the diagonal flow gives integral matrices $(Q,P)$ satisfying $QP^T=PQ^T$ , $QQ^T+PP^T$ nonsingular, and the advertised simultaneous size bounds. The conversion between Euclidean and supremum norms only changes the stated dimension-dependent constant.

Theorem 1.4Not able to verify

The Khintchine zero-one law is not established by the supplied argument

Pages 3 and 12–19 · Theorem 3.3 and proof of Theorem 1.4 · arXiv:2606.05096v1

The quoted Kleinbock–Margulis theorem gives a Borel–Cantelli conclusion for Haar-almost every point of $\Gamma\backslash\mathrm{Sp}_{2d}(\mathbb R)$ . The desired points are the fixed submanifold $\{\Gamma u_X:X\in\operatorname{Sym}_d(\mathbb R)\},$ which has ambient Haar measure zero. The proof's final paragraph simply substitutes $x=\Gamma u_X$ and concludes a statement for Lebesgue-almost every $X$ ; no local-product, stable-leaf, or expanding-horospherical transfer result is stated or proved. Such transfer theorems can be available, but their hypotheses and the needed monotonicity estimates must be verified here before the theorem follows.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 1.4 has a nontrivial missing measure-transfer step. Two additional one-symbol or one-word defects in quoted or auxiliary arguments have direct corrections and do not independently affect the overall status.

Proof of Theorem 1.4Incomplete as written

Ambient Borel–Cantelli is applied on a measure-zero horosphere

Page 19 · final proof of Theorem 1.4 · arXiv:2606.05096v1

Distance-likeness of $\Delta$ controls Haar measure of cusp targets, but Theorem 3.3 only asserts recurrence for Haar-almost every initial lattice. The map $X\mapsto\Gamma u_X$ parametrizes an unstable horosphere, not a full-measure subset. To repair the proof one needs a dynamical Borel–Cantelli theorem specifically for this horospherical parameterization, or a local-product argument showing that the target property is transferred along stable leaves with the necessary uniform error control. No such argument appears.

Corollary 3.15Minor formal correction

The action side is reversed in the homogeneity justification

Page 18 · proof of Corollary 3.15 · arXiv:2606.05096v1

Under the identification $V^2\simeq N\backslash G$ given by $Ng\mapsto Fg$ , right multiplication by $a_\varepsilon=\operatorname{diag}(\varepsilon^{-1}I_d,\varepsilon I_d)$ sends a general frame $(Q,P)$ to $(\varepsilon^{-1}Q,\varepsilon P)$ , not to $\varepsilon(Q,P)$ . Scalar dilation instead corresponds to the left map $Ng\mapsto Na_\varepsilon g$ , since $Fa_\varepsilon=\varepsilon F$ . Replacing “right” by this left action, together with the Iwasawa Jacobian already computed in Lemma 3.4, gives the claimed homogeneity $m_{V^2}(\varepsilon D)=\varepsilon^{d^2+d}m_{V^2}(D).$

Theorem 3.3(b)Typo

A logarithm law is printed with a limit instead of a limsup

Page 13 · Theorem 3.3(b) · arXiv:2606.05096v1

Cusp logarithm laws of this form assert $\limsup_{t\to\infty}\frac{\Delta(x\exp(tz))}{\log t}=\frac1\varrho,$ not existence of the full limit. Deep excursions occur intermittently, so the ratio need not converge. This quoted clause is not used in the proof of Theorem 1.4, but it is a false technical statement in the manuscript.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2606.05096v1
Authors listed: Minchang Kim
Audit date: August 18, 2026

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Khintchine's Theorem for Symmetric matrices via Flows on the Space of Symplectic Lattices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Dirichlet approximation by Lagrangian frames

The Khintchine zero-one law is not established by the supplied argument

Ambient Borel–Cantelli is applied on a measure-zero horosphere

The action side is reversed in the homogeneity justification

A logarithm law is printed with a limit instead of a limsup

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