Published paper
Abstract
The paper determines, in measure, the growth rate of denominators and the approximation rate of convergents for negative continued fractions. It also reproves metric results for the arithmetic mean of their digits, including almost-everywhere lower and upper behavior.
Role in dependence graphs
Proof-critical source
On -real and -complex numbers
This paper is included only for the following marked statement:
- Theorem 3.1, first limit · arXiv:2005.04371v1 fallback PDF p. 3Gives the almost-everywhere growth/approximation limit used to make the focal difference quotient vanish.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Pinned arXiv fallback · arXiv:2005.04371v1
Hiroaki Ito. Diophantine approximation by negative continued fraction. Tokyo Journal of Mathematics 46 (2023), no. 1, 1–18. Audited against arXiv:2005.04371v1 because the publisher PDF was inaccessible.
The Project Euclid version-of-record endpoint returned an access page rather than the article PDF; the exact cited Theorem 3.1 is present in the pinned author manuscript.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The almost-everywhere zero exponential rates for negative-continuant denominators and errors, the sharper logarithmically rescaled limits in measure, and the digit-average liminf and limsup assertions all have the correct metric qualifiers. The almost-everywhere approximation-error limit in Theorem 3.1 is exactly the estimate imported by arXiv:2508.08440v2.
Theorem 3.1No issue found
The almost-everywhere and in-measure limits are correctly distinguished
Fallback PDF page 3 · Theorem 3.1 · arXiv:2005.04371v1
The theorem first proves and for almost every . It then proves the sharper -rescaled limits with constants and in measure. These modes are not conflated, and the separate digit-average liminf and limsup conclusions keep their almost-everywhere qualifiers.
Pinned arXiv fallback version 1 ↗Approximation-error limit in Theorem 3.1Correctly used downstream
The focal derivative estimate uses no stronger conclusion
Fallback PDF page 3; focal arXiv:2508.08440v2 PDF page 30
The focal proof quotes the almost-everywhere statement , meaning that the convergent error decays subexponentially in the negative-continuant index. This is one of Ito's first two almost-everywhere limits, not one of the later in-measure refinements, so no subsequence extraction or upgrade of convergence mode is required.
02Proofs2 reported findingsNo proof issues found
The recoding of negative continued fractions as return blocks of the regular continued-fraction map, the metric estimates for those blocks, and the approximation-error identities close the proof. The pinned fallback contains the full proof of the cited theorem.
Recoding argumentNo issue found in the reviewed proof
Negative digits are correctly converted into regular continued-fraction blocks
Fallback PDF pages 3–6 · arXiv:2005.04371v1
Runs of the digit in the negative expansion correspond to return blocks in the regular expansion. The displayed continuant identities correctly relate the negative convergent denominator and error to the regular convergents at those return times. The exceptional rational and terminating cases form a null set and are excluded from the theorem's almost-everywhere scope.
Pinned arXiv fallback version 1 ↗Metric limit stepNo issue found in the reviewed proof
The classical metric theorem is applied at the correct normalization
Fallback PDF pages 5–8 · arXiv:2005.04371v1
The regular continued-fraction logarithmic denominator law and the return-time estimate are combined before normalizing by the negative index. The constants and signs in the logarithm of agree with the continuant determinant formula, giving the printed limits.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.