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arXiv:1304.4842v1

A note on Diophatine approximation in SL2(R)\rm{SL}_2(\mathbb{R})

Nikolay Moshchevitin

math.NTmath.DS11J7011H56
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Paper metadata

Original submission
April 17, 2013
Claimed MSC
11J70, 11H56
arXiv categories
math.NT, math.DS
Comments
5 pages

Abstract

We prove a quantitative version of the following statement: the unipotent flow orbit of a typical lattice in SL2(R)/SL2(Z)\rm{SL}_2(\mathbb{R})/\rm{SL}_2(\mathbb{Z}) is dense. Our quantitative result uses A. Weil's bounds for Kloostermann sums.

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Audit summary

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 20, 2026
01Statements2 reported findingsContains wrong statements

The paper's claimed admissible-exponent assertion is false as written, and that assertion is the unproved input to both main theorems. The main conclusions may admit a repaired formulation, but this audit could not verify one from the paper.

Admissible exponent assertionIncorrect

The determinant-one approximation claim has a direct counterexample

Section 1 · equations (1)-(3), definition of admissible rr · arXiv:1304.4842v1

Take ρ=1\rho=1, A1=QA_1=Q, and B1=2QB_1=2Q, so the prescribed center is (Q,−Q,2Q,−2Q)(Q,-Q,2Q,-2Q). For any fixed r<1r<1, the box (3) forces xy−zw=3Q2+O(Q1+r+Q2r)xy-zw=3Q^2+O(Q^{1+r}+Q^{2r}). This cannot equal 11 for all sufficiently large QQ. Thus the assertion that every r>3/4r>3/4 has the stated property is false with the displayed pairing and signs.

Full paper, version 1 ↗
Theorems 1 and 2Not able to verify

The orbit-approximation conclusions are not verified from the supplied argument

Sections 1 and 4 · Theorems 1-2 and their proofs · arXiv:1304.4842v1

Both proofs choose the basis vectors furnished by Lemma 2, whose only existence argument invokes the false admissible-exponent assertion. A rearrangement of the four target coordinates may be intended, but the required sign cases and the resulting basis estimates are not written out, so no complete repair of either theorem was verified.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The lattice estimate after a suitable basis is coherent, but the construction of that basis is based on a false four-variable approximation claim and Section 5 does not prove the claim actually stated.

Lemma 2Incorrect or incomplete

The required lattice basis is not established

Section 3 · Lemma 2 and equations (8)-(10) · arXiv:1304.4842v1

The proof maps two desired lattice points to integer columns (x,z)(x,z) and (y,w)(y,w) and then invokes xy−zw=1xy-zw=1 inside the box (3). Since the box claim is false, the determinant-one basis conclusion does not follow. The use of ρ=∣η2/η1∣\rho=|\eta_2/\eta_1| also discards the sign needed to justify the displayed relation between both target columns.

Section 5Incorrect as written

The Kloosterman argument targets a different coordinate arrangement

Section 5 · proof that r>3/4r>3/4 is admissible · arXiv:1304.4842v1

Section 5 chooses w=pw=p near a positive prime and then selects x,yx,y with xy≡1(modp)xy\equiv1\pmod p, defining z=(xy−1)/pz=(xy-1)/p. In the stated box, however, ww must be near B2=−ρB1B_2=-\rho B_1 while zz must be near B1B_1. For the counterexample above these targets have opposite signs, and the modular construction cannot satisfy them.

Lemma 3Correct

The conditional matrix estimate is algebraically sound

Section 3 · Lemma 3 · arXiv:1304.4842v1

Assuming a determinant-one pair with the stated horizontal and vertical bounds, the explicit multiplication by the two unimodular matrices yields the claimed entrywise estimates. This part does not repair the missing basis existence input.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1304.4842v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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