2404.09424
POLYNOMIAL FOURIER DECAY FOR PATTERSON-SULLIVAN MEASURES
Osama Khalil
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The uploaded paper proves a polynomial nonstationary-phase bound for Patterson–Sullivan (PS) measures (Theorem 1.1) by reducing to linear phases along N^+-leaves (Lemma 3.1), performing an explicit holonomy linearization (Lemma 3.2 and Corollary 3.3), then combining Cauchy–Schwarz, frequency separation, and a counting argument with an L^2-flattening theorem for uniformly affinely non-concentrated measures (Theorem 2.3, Proposition 2.4), yielding the claimed |λ|^{-κ} decay with κ depending only on the PS non-concentration parameters; the constant C depends on A, a, µ. These ingredients are all present in the paper and clearly align with the candidate’s outline: reduction to leafwise integrals and flow boxes, linearization via stable holonomy, application of L^2-flattening to leafwise PS conditionals, and summation/exceptional set control. Theorem 1.1 and its hypotheses match exactly the candidate’s statement, including the dependence of constants and the gradient lower bound on φ, and the proof’s structure matches the candidate’s steps, down to the holonomy formulas and the exceptional-set counting. See Theorem 1.1 for the exact statement and constant dependence, and its discussion of dependence on non-concentration parameters (κ) and on A,a,µ (C) ; the reduction to linear phases (Lemma 3.1, eq. (3.1)) ; the L^2-flattening framework and uniform affine non-concentration (Definition 2.2, Theorem 2.3, Proposition 2.4) ; the Cauchy–Schwarz and time-averaging step (equation (3.23)) ; separation of frequencies and lower bounds for β_{j,k,ℓ} (the “Separation of frequencies” display following (3.29) and the matrix estimate) ; and the explicit stable holonomy formula (Section 4) used in the linearization . Minor presentational differences (e.g., the smoothing window [T,2T] versus (T−1,T)) are immaterial to the method and constants. Hence both the paper and the model present substantially the same, correct proof.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} strong field \textbf{Justification:} The paper presents a concise, robust proof of polynomial Fourier decay for Patterson–Sullivan measures in the convex cocompact setting, leveraging an L\^2-flattening theorem together with explicit holonomy linearization and frequency separation. The result generalizes prior dimension-specific works and has significant downstream implications (e.g., essential spectral gaps). The presentation is clear and modular; minor expansions (e.g., the proof of the reduction lemma) would improve readability.