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Paper metadata

Original submission
March 25, 2013
Claimed MSC
11J13
arXiv categories
math.NT
Comments
6 pages

Abstract

We prove a result on approximations to a real number θθ by algebraic numbers of degree ≤2\le 2 in the case when we have information about the uniform Diophantine exponent ω^\hatω for the linear form x0+θx1+θ2x2x_0 +θx_1+θ^2x_2.

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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 20, 2026
01Statements2 reported findingsCorrect

The two-linear-form exponent bound and its quadratic-approximation corollary are correct; the final recurrence calculation in the printed proof needs a verified repair.

Theorem 1Correct · verified repair

Relative small-value exponent bound

Pages 1 and 5-6 · Theorem 1 and Section 5 · arXiv:1303.6093v1

Lemmas 3-5 give the stated bound after correcting the recurrence limit described below. The exact repaired fixed point still forces every r<α2−α+1r<\alpha^2-\alpha+1 to fail on infinitely many minimal points. Letting α↑ω^\alpha\uparrow\widehat\omega yields ωLP≥ω^2−ω^+1\omega_{LP}\geq\widehat\omega^2-\widehat\omega+1.

Full paper, version 1 ↗
Theorem 2Correct

Quadratic-algebraic approximation corollary

Pages 1-2 · Theorem 2 and inequality (3) · arXiv:1303.6093v1

For L(x)=x0+x1θ+x2θ2L(x)=x_0+x_1\theta+x_2\theta^2 and P(x)=x1+2x2θP(x)=x_1+2x_2\theta, the standard root perturbation estimate converts ∣L(x)∣≤∣P(x)∣H−r|L(x)|\leq|P(x)|H^{-r} into approximation of θ\theta by a degree-at-most-two root with exponent rr. Theorem 1 therefore gives (4).

02Proofs3 reported findingsContains incorrect or incomplete proofs

The determinant lemmas are correct, but the final recurrence has a false closed form as printed. A verified algebraic repair preserves both theorems.

Lemma 5Correct and complete after local notation repairs

The main determinant estimate is correct

Pages 4-5 · Lemma 5 and equations (15)-(22) · arXiv:1303.6093v1

Lemma 4 makes the first term of the two-by-two determinant dominate. The endpoint hypotheses bound the other side by Lk−1LkXkrL_{k-1}L_kX_k^r, while Jarnik growth converts XkX_k to XνX_\nu. This gives (19) and the improved lower exponent β′=r−α−1+β0/(α−1)\beta'=r-\alpha-1+\beta_0/(\alpha-1).

Section 5 recurrenceIncorrect as written · verified repair

The displayed solution of the affine recurrence is false

Page 5 · proof of Theorem 1, recurrence for βi\beta_i · arXiv:1303.6093v1

For βi+1=r−α−1+βi/(α−1)\beta_{i+1}=r-\alpha-1+\beta_i/(\alpha-1), the printed limit α(α−1)\alpha(\alpha-1) is not the fixed point unless r=α2−α+1r=\alpha^2-\alpha+1. The correct fixed point is B=(r−α−1)(α−1)/(α−2)B=(r-\alpha-1)(\alpha-1)/(\alpha-2), and βi=B+(β0−B)/(α−1)i\beta_i=B+(\beta_0-B)/(\alpha-1)^i. Applying Lemma 5 in the limit gives r≥α2+1−B/(α−1)r\geq\alpha^2+1-B/(\alpha-1), which rearranges to r≥α2−α+1r\geq\alpha^2-\alpha+1. This contradicts (12) and completes a verified repair without changing the theorem.

Equations (11), (18), and final Lemma 5 useTypos

Three local symbols require mechanical correction

Pages 3-5 · equations (11), (18), and final application of (19) · arXiv:1303.6093v1

The determinant recurrence in (11) has a minus rather than a plus, although only its absolute-value triangle bound is used. In (18), XνrX_\nu^r must be XkrX_k^r, as the proof immediately uses. In the last application of (19), the displayed bound is α2+1−βw/(α−1)\alpha^2+1-\beta_w/(\alpha-1), not α2−α+1−βw/(α−1)\alpha^2-\alpha+1-\beta_w/(\alpha-1).

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1303.6093v1
Authors listed
Nikloay Moshchevitin
Audit date
August 20, 2026
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