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arXiv:2401.15586v2

On the rate of convergence of continued fraction statistics of random rationals

Ofir David, Taehyeong Kim, Ron Mor, Uri Shapira

math.DSmath.NT37A3537A4411J70
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Paper metadata

Original submission
January 28, 2024
Latest revision
November 17, 2024 · version 2
Claimed MSC
37A35, 37A44, 11J70
arXiv categories
math.DS, math.NT

Abstract

We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator qq, approaches the Gauss-Kuzmin statistics with polynomial rate in qq. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator qq and prime numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space SL2(R)/SL2(Z)SL_2(\mathbb{R})/SL_2(\mathbb{Z}).

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Audit summary

Audited against arXiv v2

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.

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Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The quantitative continued-fraction equidistribution results, their stability for large denominator subsets, the prime-numerator consequence, and the non-escape-of-mass theorem are correct.

Theorems 1.2–1.4Correct

The continued-fraction estimates are correct

Pages 2–5 and 15–19 · Theorems 1.2–1.4 · arXiv:2401.15586v2

The geodesic-orbit estimates translate to Gauss-word frequencies and normalized continued-fraction lengths with the displayed constants. The exceptional-set count is polynomially smaller than the denominator family, so any subset of size q1−o(1)q^{1-o(1)} inherits the convergence, and the prime-numerator family satisfies that size condition.

Full paper, version 2 ↗
Theorem 1.6 and Corollary 1.7Correct

The mass and equidistribution conclusions follow with the stated threshold

Pages 6–7 and 10–15 · Theorem 1.6 and Corollary 1.7 · arXiv:2401.15586v2

The covering estimate bounds the proportion of starting points whose trajectories spend at least a fixed fraction outside a compact set. Averaging over a denominator subset of exponent 2h2h leaves at least 2h−12h-1 mass in the space. When h=1h=1, entropy rigidity identifies the full weak-star limit with Haar measure.

02Proofs2 reported findingsCorrect

The escape estimate, entropy argument, and continued-fraction deductions are correct and complete. One sentence reverses ‘upper’ and ‘lower’ but the displayed inequality and its use are correct.

Proof of Theorem 1.6Typo

An upper bound is called a lower bound

Page 13 · sentence immediately after equation (2.2) · arXiv:2401.15586v2

Equation (2.2) bounds the escaping mass δΛq[0,log⁡q](X>M)\delta_{\Lambda_q}^{[0,\log q]}(X^{>M}) from above, but the following sentence calls it a ‘lower bound’. The next line correctly subtracts it from 11 to obtain a lower bound for compact mass. Replacing ‘lower’ by ‘upper’ is the unique correction and changes no inference.

Sections 2–3Correct and complete

The orbit-covering and continued-fraction steps close

Pages 10–19 · Sections 2–3 · arXiv:2401.15586v2

The continuous-time visit proportion is compared in the correct direction with the sampled flow, the covering exponent is combined with the subset cardinality before limits are taken, and the coding identities transfer the result to word frequencies and lengths without losing the stated error rate.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2401.15586v2
Authors listed
Ofir David, Taehyeong Kim, Ron Mor, Uri Shapira
Audit date
August 19, 2026
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