MathAudit

Paper metadata

Original submission
December 12, 2009
Latest revision
September 6, 2010 · version 3
Claimed MSC
11J13
arXiv categories
math.NT
Comments
The results of the second version are better than that from the first one. Misprints corrected in the third version

Abstract

We find new inequalities between uniform and individual Diophantine exponents for three-dimensional Diophantine approximations. Also we give a result for two linear forms in two variables. The results improves V.Jarnik's theorem (1954).

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Audit summary

Audited against arXiv v3

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
01Statements1 reported findingCorrect

The three exponent improvements are correct; the detected defects are uniquely repairable notation typos.

Theorems 2–4Correct

Exponent bounds in the three-dimensional and 2×22\times2 cases

Pages 2–3 · Theorems 2, 3, and 4 · arXiv:0912.2442v3

The three case analyses give respectively β≥αg1(α)\beta\geq\alpha g_1(\alpha), β≥αg2(α)\beta\geq\alpha g_2(\alpha), and β≥αg3(α)\beta\geq\alpha g_3(\alpha). In each proof the defining polynomial identity for gig_i makes the exponents from all exhaustive growth alternatives agree.

02Proofs5 reported findingsCorrect

After the mechanical corrections listed below, the determinant estimates, lattice comparisons, and growth alternatives form complete proofs.

Equation (4)Typo

The system's left-hand variable is δ\delta, not γ\gamma

Page 2 · Equation (4) · arXiv:0912.2442v3

Replace the first printed γ\gamma by δ\delta, so that Equation (4) reads δ=1α+α−1αδγ=αγ(1−α)−α\delta=\frac1\alpha+\frac{\alpha-1}{\alpha}\frac{\delta}{\gamma}=\frac{\alpha}{\gamma(1-\alpha)-\alpha}. The three cases in Section 4 require exactly these two identities, and eliminating γ\gamma gives the displayed quadratic whose largest root is δ=g1(α)\delta=g_1(\alpha). The correction is unique and restores every cited use.

Proof of Theorem 2Typo

The concluding equation references should be (10), (11), and (12)

Page 5 · final paragraph of Section 4 · arXiv:0912.2442v3

The proof prints that Theorem 2 follows from (10), (15), and (18), but Equations (15) and (18) occur only in the later proof of Theorem 4. Replace them by (11) and (12). Those are precisely the second and third alternatives just proved in Section 4.

Proof of Theorem 3Correct and complete

The best-approximation block comparison is valid

Pages 5–7 · Equation (13), Lemma 2, and proof of Theorem 3 · arXiv:0912.2442v3

If the best approximations are eventually three-dimensional, Jarník's m=2m=2 bound is stronger. Otherwise the four-dimensional determinant and the induced two-dimensional lattice give three exhaustive growth regimes. The identity αg2(α)2+(α−2)g2(α)−(α−1)2=0\alpha g_2(\alpha)^2+(\alpha-2)g_2(\alpha)-(\alpha-1)^2=0 matches the exponent in the intermediate regime.

Proof of Theorem 4Typo

The third determinant row has a wrong final coordinate

Page 8 · determinant at the start of case R(Θ)=4R(\Theta)=4 · arXiv:0912.2442v3

Replace the final entry x2,kx_{2,k} in the third row by y2,ky_{2,k}. The row is the best-approximation vector (x1,k,x2,k,y1,k,y2,k)(x_{1,k},x_{2,k},y_{1,k},y_{2,k}), as fixed immediately above, and the determinant estimate that follows uses that vector. The correction is mechanical and preserves the proof.

End of Section 6Typo

The final theorem number is 4

Page 8 · last sentence of the proof · arXiv:0912.2442v3

Replace “Theorem 2 follows” by “Theorem 4 follows.” Section 6 is the proof of Theorem 4 and uses g3g_3, so the intended reference is uniquely determined.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0912.2442v3
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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