Published paper
Abstract
Proposition 6.2 and its internal combinatorial proof are the cumulant input used in the focal central-limit argument.
Role in dependence graphs
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:
- Proposition 6.2 · arXiv v1 PDF p. 12; proof §§10.1–10.2, PDF pp. 17–18Exhausts tuple configuration space by clustered and partition-separated pieces.
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Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:1706.09167v1 · explicit fallback for the inaccessible Journal d’Analyse Mathématique version of record
Michael Björklund, Alexander Gorodnik. Central limit theorems for group actions which are exponentially mixing of all orders. Journal d’Analyse Mathématique 141 (2020), no. 2, 457–482. Reviewed in the exact arXiv:1706.09167v1 form.
The publisher version of record was not publicly downloadable during this audit, and no independently authenticated author manuscript matching the journal pagination was available. The complete arXiv version 1 was reviewed as the explicit open proof-bearing fallback; this report does not claim inspection of the publisher file.
Open audited source ↗01Statements3 reported findingsCorrect
The central-limit theorem for Følner averages of actions that are exponentially mixing of all orders, the general measure-averaging criterion, and the homogeneous and lacunary-unipotent applications are correct. One variance display has a uniquely determined normalization typo.
Theorem 1.1 and Corollary 1.3Correct
Subexponential-growth averages satisfy the asserted central-limit theorem
PDF pages 3–4 and Sections 4–5 · Theorem 1.1, Corollary 1.3, and proofs · arXiv:1706.09167v1
For a right Følner sequence in a group of subexponential metric-volume growth, the normalized Haar measures satisfy the local-mass condition of Theorem 1.5. Exponential two-point mixing makes the covariance integrable, and the Følner property identifies the limiting variance as . The cumulant theorem then gives convergence to the centered Gaussian with this variance. The Cartan-action corollary uses the stated strong-spectral-gap input to supply exponential mixing of all orders.
Exact arXiv version 1 ↗Theorem 1.5Typo
The general central-limit criterion is correct after a variance-notation correction
PDF page 4 · equation (1.7) in Theorem 1.5 · arXiv:1706.09167v1
The theorem prints and then denotes the limiting Gaussian by . The unique correction, confirmed by the proof and Proposition 5.1, is . With that correction, the local-mass hypothesis kills every cumulant of order at least three and the assumed second-moment limit is exactly the Gaussian variance.
Exact arXiv version 1 ↗Corollary 1.6Correct
Lacunary sampling of a unipotent flow has the stated Gaussian limit
PDF pages 4–5 and Section 5 · Corollary 1.6 and proof · arXiv:1706.09167v1
For the normalized counting measures on lacunary times, any logarithmic metric ball contains only a uniformly bounded number of sample points relative to the required normalization. Polynomial multiple mixing of unipotent translates becomes exponential in the sample index because the times are lacunary. Distinct off-diagonal correlations vanish in the variance computation, leaving , as claimed.
Exact arXiv version 1 ↗02Proofs3 reported findingsCorrect
The cumulant decomposition, separation estimates, local-mass argument, and application proofs are correct and complete. Proposition 6.2—the combinatorial decomposition used by the focal paper—is proved directly. One omitted total-mass factor in an intermediate displayed bound is a mechanical typo; the proposition statement and its application contain the factor.
Theorem 1.5 proofCorrect and complete
Clustered and separated tuples give vanishing higher cumulants
PDF pages 9–15 · Sections 5–9 and proof of Theorem 1.5 · arXiv:1706.09167v1
The joint cumulant is integrated over . Tuples inside a logarithmic cluster are controlled by the assumed local mass, while tuples split into well-separated blocks have exponentially small conditional cumulants by multiple mixing and the seminorm growth estimates. The recursively chosen separation scales make both contributions tend to zero. Proposition 5.1 then converts vanishing cumulants and convergence of the second moment into the claimed Gaussian limit.
Exact arXiv version 1 ↗Proposition 6.2Correct and complete
The finite cluster decomposition of is valid
PDF page 12 and PDF pages 18–20 · Proposition 6.2 and Section 10 proof · arXiv:1706.09167v1
Starting from the discrete partition of , if two blocks are not separated at the next threshold the proof merges them. The triangle inequality gives diameter below three times that threshold for the merged block. Each failure therefore produces a strictly coarser partition, so after at most mergers either the whole tuple is clustered or it belongs to one of the announced separated-block regions. This proves the exact covering used in the cumulant estimate and in the focal argument.
Exact arXiv version 1 ↗Proof of Proposition 5.2Typo
The last intermediate estimate omits the measure’s total-mass factor
PDF page 12 · final display in the proof of Proposition 5.2 · arXiv:1706.09167v1
After bounding the conditional cumulant uniformly by , the integral over a separated region is printed with the same bound but without its measure. The unique correction is to multiply that term by , since the region has -measure at most . Proposition 5.2 itself includes this factor, and the next proof substitutes the corrected stated estimate, so the argument is unchanged.
Exact arXiv version 1 ↗03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.