Published paper
Abstract
Theorem 1.1 gives the explicit one-parameter rank-one nondivergence estimate that Shi integrates in polar coordinates in the proof of Lemma 3.3.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Theorem 1.1 · source p. 2; proof §§3–4Explicit one-parameter unipotent nondivergence on a rank-one quotient; Shi applies its explicit calculation in (3.10) and integrates in polar coordinates.
Proof-critical source
Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation
This paper is included only for the following marked statement:
- Theorem 1.1 · source p. 2; proof §§3–4Explicit rank-one unipotent nondivergence with the dependence Shi integrates in polar coordinates.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:1408.2591v1 · explicit fallback for the inaccessible ETDS version of record
C. Davis Buenger and Cheng Zheng. Non-divergence of unipotent flows on quotients of rank-one semisimple groups. Ergodic Theory and Dynamical Systems 37 (2017), no. 1, 103–128. Reviewed in arXiv:1408.2591v1 because the Cambridge version of record was not publicly downloadable.
The exact Cambridge version of record was paywalled during this audit. The sole public arXiv version was reviewed as the explicit proof-bearing fallback; this report does not claim inspection of the publisher file.
Open audited source ↗01Statements4 reported findingsContains unsupported statements
The rank-one quantitative nondivergence theorem, its uniform obstruction dichotomy, and the product theorem under the stated projection hypothesis are supported after the recorded repairs. The public arXiv fallback also states an extension to every discrete subgroup of a product of factors, but supplies no argument that removes the projection hypothesis; that extension is therefore unsupported in the reviewed form.
Theorems 1.1 and 1.3Correct
The rank-one quantitative and uniform conclusions have the advertised dependence
arXiv:1408.2591v1, PDF pages 2–3 and Sections 3–4
Theorem 1.1 obtains a power-law bound for the time spent outside a prescribed compact set, with constants depending only on the ambient rank-one group and the requested error. Theorem 1.3 makes the compact set uniform over a compact family of initial conditions unless a fixed unipotent-generated abelian subgroup remains uniformly small. The good-function estimate and nilpotent-hull reduction provide precisely these alternatives after the hull-selection repair recorded in the proof audit.
arXiv:1408.2591v1 ↗Theorem 1.4Correct
The product theorem matches the subgroup classification after one wording correction
arXiv:1408.2591v1, PDF pages 3 and 13–23 · Theorem 1.4 and Section 5
For products of rank-one groups, the proof needs every nontrivial coordinate projection of a lattice element to be non-elliptic; a projection equal to the identity is allowed. The displayed property in Section 5 should therefore read “either trivial or non-elliptic.” With that correction, Lemma 5.1 and the induction by the number of nontrivial coordinates apply exactly under Theorem 1.4's stated projection hypothesis.
arXiv:1408.2591v1 ↗Theorem 1.5Not able to verify
The arbitrary-discrete-subgroup extension is not proved in the reviewed fallback
arXiv:1408.2591v1, PDF page 3 and Section 5
Theorem 1.5 asserts the conclusions for every discrete subgroup of a product of factors. Section 5, however, assumes property throughout and proves only Theorem 1.4 under the corresponding no-nontrivial-elliptic-projection hypothesis. The observation that a single noncentral elliptic element has a controlled centralizer does not establish the nilpotent-subgroup classification and overlap induction for arbitrary elliptic projections. No argument removing appears in the sole public arXiv source, so this stronger statement is unsupported in the reviewed fallback; this report makes no claim about whether the inaccessible publisher text repairs or narrows it.
arXiv:1408.2591v1 ↗Theorem 1.1 as used by ShiCorrect
The one-parameter slice estimate supplies Shi’s required power law
arXiv:1408.2591v1, PDF pages 2 and 11 · Theorem 1.1 and its proof
The theorem bounds the bad-time set by once the orbit segment contains one point with no sufficiently small stabilizer element. Its exponent and compact-set dependence are uniform in the initial point. This is exactly the estimate that Shi applies to radial one-parameter slices before integrating in polar coordinates; none of the product-only formal defects enter that use.
arXiv:1408.2591v1 ↗02Proofs4 reported findingsContains incorrect or incomplete proofs
The quantitative mechanism is sound, but Lemma 3.3 incorrectly treats independently selected finite-prefix nilpotent hulls as nested. A standard finite-generation and uniform-Zassenhaus argument repairs that lemma. The public fallback gives no proof of Theorem 1.5. The remaining defects are mechanical constant, hypothesis-wording, and indexing errors.
Lemma 3.3Incomplete as written · verified repair
The finite-prefix nilpotent hulls are not nested by construction
arXiv:1408.2591v1, PDF page 6 · Lemma 3.3
For each finite prefix of a discrete subgroup, the proof independently chooses a connected nilpotent Lie subgroup containing that prefix and then says the resulting Lie algebras are increasing “by construction.” Nothing in the choices enforces nesting, so their union need not be a Lie algebra. The repair uses Proposition 4 of Goldman–Hirsch's generalized Bieberbach theorem: for a discrete subgroup of a compact-by-connected-nilpotent Lie group, the subgroup generated by elements with compact projection in a sufficiently small fixed neighborhood is finite-index, nilpotent, and finitely generated. Choose a finite subset of the paper's set that generates this subgroup, contract those finitely many elements simultaneously into the Zassenhaus neighborhood, place them in one connected nilpotent subgroup, and apply the inverse automorphism. This yields the single connected nilpotent hull required by Lemma 3.3 and preserves its uniform index bound.
Goldman–Hirsch, Proposition 4 · exact public version of record ↗Theorem 1.5Incomplete as written
No proof removes the product-section ellipticity hypothesis
arXiv:1408.2591v1, PDF page 3 and Section 5
The product proofs begin by imposing property and use it in Lemma 5.1 and again when comparing dominant subgroups. The preprint never supplies the separate classification or modified overlap argument required when a coordinate projection is a nontrivial elliptic element. Consequently the proof establishes Theorem 1.4, after replacing by “either trivial or non-elliptic,” but does not establish the stronger Theorem 1.5 in the reviewed arXiv fallback.
arXiv:1408.2591v1 ↗Proof of Theorem 1.1Typo
The final constant assignment overwrites the fixed threshold
arXiv:1408.2591v1, PDF page 11 · final paragraph of the proof of Theorem 1.1
After fixing , choosing from , and defining the compact set through a constant , the proof says “let .” This would overwrite the already fixed threshold and contradict the displayed value of . The unique correction is , which gives the printed formula .
arXiv:1408.2591v1 ↗Section 5Typos
Two product-proof indexing statements need mechanical corrections
arXiv:1408.2591v1, PDF pages 13 and 17 · property and Proposition 5.4
First, property says every coordinate projection of a nonidentity element is non-elliptic, whereas Theorem 1.4 and the proof allow that projection to be the identity; insert “either trivial or.” Second, the backward induction in Proposition 5.4 begins “,” which omits the step needed to construct and ; replace it by . Both corrections are uniquely forced by the adjacent formulas.
arXiv:1408.2591v1 ↗03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.