MathAudit

Paper metadata

Original submission
September 30, 2013
Claimed MSC
11J13
arXiv categories
math.NT
Comments
7 pages

Abstract

We obtain some new inequalities between the ordinary and the uniform Diophantine exponents for simultaneous Diophantine approximation to four real numbers.

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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
01Statements3 reported findingsContains unsupported statements

The index-one bound is supported. Every index-two or index-three branch depends on a cone lemma whose required normalization or decisive computation is absent, so those branches could not be independently verified from the supplied details.

Theorem 1(1)Correct

The case of index one is verified

Sections 2 and 4 · Theorem 1(1) and case i(Θ)=1\mathfrak i(\Theta)=1 · arXiv:1309.7826v1

The nonzero five-by-five determinant gives the growth alternative (11). Substituting the uniform approximation bounds and balancing its two possible dominant terms gives the polynomial defining G1(ω^)G_1(\widehat\omega), hence ω≥ω^G1(ω^)\omega\geq\widehat\omega G_1(\widehat\omega).

Theorem 1(2), first branchNot able to verify

The branch ω^≤1/2\widehat\omega\leq1/2 is not established by Lemma 1 as written

Section 5 · system (17), Lemma 1, and the first part of (9) · arXiv:1309.7826v1

The paper displays the fifteen-variable system and its facet matrix, but it does not supply the ray calculation that would yield G2,1G_{2,1}. More basically, Lemma 1 allows X=0\mathfrak X=0, which satisfies its homogeneous hypotheses for every g>0g>0 although the asserted conclusion fails when g<G2,1(α)g<G_{2,1}(\alpha). The later logarithmic vector also has Xν=log⁡qν>0X_\nu=\log q_\nu>0 and is never shown to lie in the stated subspace R′={Xν=0}\mathcal R'=\{X_\nu=0\}. A nonzero normalization and a corrected cone argument may repair the branch, but neither is written or independently verified here.

Theorem 1(2)-(3), remaining branchesNot able to verify

The high branch and index-three bound are not fully demonstrated

Sections 6-7 · Lemmas 2-3 and the second part of (9) and (10) · arXiv:1309.7826v1

The paper explicitly calls these arguments sketches. Lemma 2 is said to have a similar proof without the required cone calculation, and Lemma 3 is followed only by the instruction to consider a nineteen-dimensional cone. Those omitted computations are precisely what must exclude all other vertices and establish the displayed roots G2,2G_{2,2} and G3G_3.

02Proofs4 reported findingsContains unverified proofs

The determinant proof of the index-one regime is complete. All index-two and index-three regimes rely on cone arguments whose normalization or decisive extremal-ray computations are absent. Several branch labels contain uniquely repairable typographical slips.

Section 4 determinant argumentCorrect and complete

The index-one growth dichotomy is complete

Section 4 · equations (11)-(16) · arXiv:1309.7826v1

Linear independence makes the integral determinant nonzero. Expanding in approximation errors supplies (12); the two assumed upper growth bounds then contradict it unless at least one adjacent denominator grows with the exponent dictated by G1G_1.

Lemma 1Incomplete as written

The normalized-cone implication is incomplete as stated

Section 5 · Lemma 1, system (17), matrix G\mathfrak G, and equations (18)-(19) · arXiv:1309.7826v1

Invertibility of the displayed facet matrix can show that the chosen homogeneous cone is simplicial, but it does not by itself prove the claimed intersection threshold. The zero vector is a formal counterexample to Lemma 1 as stated, and the proof neither imposes a nonzero normalization nor lists the extremal rays or an equivalent dual certificate. Its application also does not justify replacing Xν=log⁡qνX_\nu=\log q_\nu by zero.

Lemmas 2 and 3Incomplete

The decisive cone analyses are omitted

Sections 6-7 · Lemmas 2-3 · arXiv:1309.7826v1

No vertices, inverse matrix, or dual certificate are supplied for either altered inequality system. Because a changed active inequality can create new extremal rays, similarity to Lemma 1 does not formally imply the two stated polynomial thresholds.

Branch labels and comparison chainTypos · no status impact

Local branch labels and symbols are typographical slips

Sections 2 and 5-7 · index discussion, proof of Lemma 1, branch conclusions, and system (22) · arXiv:1309.7826v1

The index discussion repeats θ1\theta_1 where the fourth component must be θ4\theta_4. The proof of Lemma 1 refers to (23) where it must refer to its conclusion (19). In the low branch, xk+1x_{k+1} must be Xk+1X_{k+1}, ∣log⁡ξj∣|\log\xi_j| must be ∣ξj∣|\xi_j|, and the concluding exponent must be G2,1G_{2,1} rather than G2,2G_{2,2}. The high-branch sketch must invoke Lemma 2 rather than Lemma 1. Finally, system (22) is missing the comparison sign between ξr2\xi_{r_2} and ξr3−1\xi_{r_3-1}. Each correction is forced by the branch definitions and the surrounding parallel systems.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1309.7826v1
Authors listed
Dmitry Gayfulin, Nikolay Moshchevitin
Audit date
August 20, 2026
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