MathAudit

Paper metadata

Original submission
March 20, 2007
Claimed MSC
11J70
arXiv categories
math.NT
Comments
27 pages, 2 figures, submitted to Acta Arithmetica

Abstract

We generalize the Farey-Brocot partition to a twodimensional continued fraction algorithm and generalized Farey-Brocot nets. We give an asymptotic formula for the moments of order β.

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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 findingsCorrect

The global convergence result for Algorithm A and the two moment asymptotics are correct after verified finite-index and denominator-bound repairs.

Theorem 1Correct

Global weak convergence of Algorithm A

Pages 12–13 · Theorem 1 · arXiv:math/0703566v1

The denominator ratio on every retained edge lies between 1/ν1/\nu and 1−1/ν1-1/\nu. Each refinement therefore contracts the containing triangle's diameter by at most 1−1/ν1-1/\nu, and the product of these factors tends to zero. This proves weak convergence at every point.

Theorems 2 and 3Correct

Moment asymptotics for Algorithms A and B

Pages 18 and 24–25 · Theorems 2 and 3 · arXiv:math/0703566v1

The code decompositions isolate the final long run as the main term and bound all other codes by convergent Dirichlet-series tails. In Algorithm B, replacing the false pointwise denominator bound in Lemma 18 by the corrected n+O(w)n+O(w) bound changes only the existing relative O(w/n)O(w/n) term, so the leading constant and stated error remain unchanged.

02Proofs3 reported findingsContains incorrect or incomplete proofs

Several displayed inequalities in the moment estimates are false at small indices or omit an O(w)O(w) denominator term. The repairs below are verified and preserve both asymptotic theorems.

Lemma 18Incorrect as written · verified repair

The lower area bound uses n/2n/2 instead of n/2+O(w)n/2+O(w)

Page 24 · proof of Lemma 18 · arXiv:math/0703566v1

After k=n−wk=n-w zero operations, the two changing denominators are q(a)+kq(c)/2q(a)+kq(c)/2 and q(b)+kq(c)/2q(b)+kq(c)/2. Lemma 8 gives q(a),q(b)≤(w+1)q(c)q(a),q(b)\leq(w+1)q(c), so their upper bound is (n/2+O(w))q(c)(n/2+O(w))q(c), not the printed (n/2)q(c)(n/2)q(c). Replacing the left sandwich term by the corresponding (n+O(w))2/2(n+O(w))^2/2 denominator yields the same main term multiplied by 1+O(w/n)1+O(w/n), exactly within Lemma 18's stated error. In the same paragraph, sum new vertices through stage ww, rather than stopping at w−1w-1, to match VwV_w; the added layer is covered by the same tail estimate.

Lemmas 9 and 17Incorrect as written · verified repair

Two finite boundary cases are used outside their displayed bounds

Pages 14 and 22–23 · Equation (zeta) and proof of Lemma 17 · arXiv:math/0703566v1

The count bounded by ((q+1)2−4)/q2≤4/3((q+1)^2-4)/q^2\leq4/3 excludes the four q=1q=1 points, and ⌊t/2⌋−1≥t/3\lfloor t/2\rfloor-1\geq t/3 fails for several small tt. Separate q=1q=1 as a finite contribution and start the latter estimate once t≥8t\geq8, absorbing the finitely many smaller blocks into the constant. Both series retain the same convergence and the asymptotic error estimates are unchanged.

Lemma 13 and Lemma 18 notationTypo

Two object names in displayed formulas are swapped

Pages 20 and 24 · Lemma 13(ii) and proof of Lemma 18 · arXiv:math/0703566v1

Close the missing parenthesis in the area calculation of Lemma 13(ii), and replace mes⁡Δ′\operatorname{mes}\Delta' by mes⁡Δ\operatorname{mes}\Delta in the formula after the zero-operation recursion. The denominators and the surrounding sentences uniquely determine both corrections.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:math/0703566v1
Authors listed
Nikolai Moshchevitin, Michael Vielhaber
Audit date
August 20, 2026
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