MathAudit

Paper metadata

Original submission
March 8, 2026
Latest revision
October 19, 2026 · version 2
Claimed MSC
37D40, 11J70, 11K50
arXiv categories
math.DS, math.NT
Comments
38 pages, 7 figures

Abstract

We establish an extreme value theorem for the geodesic flow gtg_t on the hyperbolic surface Θ\H2Θ\backslash\mathbb{H}^2 associated with the theta group ΘΘ. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd--odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross-section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on Θ\H2Θ\backslash\mathbb{H}^2.

AI-generated audit

Audit summary

Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 23, 2026
01Statements2 reported findingsNo statement issues found

The extreme-value law for spliced continued-fraction digits and its geometric cusp-excursion counterpart are correctly normalized. The cross-section, return-time constant, and conversion factor 4/π24/\pi^2 agree with the invariant measures and the digit tail. No mathematical statement requiring correction was found.

Theorem 1.1No issue found

The digit maximum has the stated Fréchet-type limit

PDF pages 3 and 18–23 · Theorem 1.1 and Section 4 · arXiv:2603.07649v2

The invariant tail satisfies μ{a1>N}∼C0/N\mu\{a_1>N\}\sim C_0/N with C0=2/log⁡(2+3)C_0=2/\log(2+\sqrt3). The spectral-gap estimate supplies exponentially decaying relative correlations for separated exceedances. Inclusion–exclusion, with close clusters and high-order tails controlled separately, then gives exp⁡(−1/y)\exp(-1/y) at threshold C0NyC_0Ny.

Theorem 1.2No issue found

The geometric height law has the correct time normalization

PDF pages 3–4 and 23–28 · Theorem 1.2 and Section 5 · arXiv:2603.07649v2

The mean return time is C∗=π2/(2log⁡(2+3))C^*=\pi^2/(2\log(2+\sqrt3)), while the digit threshold constant is C0=2/log⁡(2+3)C_0=2/\log(2+\sqrt3), so their ratio is 4/π24/\pi^2. The maximal distance differs from the logarithm of the maximal digit by errors tending to zero, and concentration of the return count transfers the digit law to the deterministic time interval.

02Proofs2 reported findingsNo proof issues found

The natural-extension coding, transfer-operator argument, exceedance estimates, and conversion from symbolic digits to geometric excursions form a complete chain. Boundary excursions, the extra terminal excursion, arbitrary absolutely continuous initial measures, and random return counts are accounted for. No incorrect or materially incomplete proof was found.

Theorems 3.5, 3.7, and 3.10No issue found in the reviewed proof

The symbolic system is matched to the geodesic return map

PDF pages 11–16 · Section 3 · arXiv:2603.07649v2

The branch matrices take each cross-section component to the component prescribed by the spliced digit, giving the natural extension of the interval map after the stated coordinate change. Integrating the roof function against the invariant cross-section measure recovers the Liouville volume and hence the displayed almost-sure return-time constant.

Lemmas 5.2–5.6No issue found in the reviewed proof

Digit size and excursion height are compared with vanishing error

PDF pages 24–28 · Section 5 · arXiv:2603.07649v2

Each spliced digit counts the quadrilaterals crossed in one excursion and differs from the exponential midpoint height by a bounded amount. Excursions in compact geometric regimes contribute only a transient bounded error, while large excursions give an exponentially small distance error. The additional (N+1)(N+1)-st excursion handles a return point already high in a cusp, and ergodicity concentrates the number of returns around T/C∗T/C^*.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2603.07649v2
Authors listed
Jaelin Kim, Seul Bee Lee, Seonhee Lim
Audit date
October 23, 2026
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