2205.12366
Dynamical Borel–Cantelli Lemma for Recurrence under Lipschitz Twists
Dmitry Kleinbock, Jiajie Zheng
correcthigh confidence
- Category
- math.DS
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper rigorously proves the zero–one law for R_f^T(ψ) under assumptions (1.7)–(1.12) via precise lower/upper bounds on μ(A_n) with additive mixing errors and a controlled pair-correlation estimate, culminating in a full-measure statement using an abstract second-moment lemma and Egorov’s theorem. By contrast, the model’s proof contains a critical over-simplification (reducing the moving center to a single center per cylinder), adds an unannounced monotonicity assumption on ψ, omits essential additive error terms from mixing, and overclaims a strong Borel–Cantelli law not established by the paper.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper establishes a unified zero–one law for twisted recurrence with a clean set of assumptions and a robust proof strategy. The arguments carefully manage geometric distortion and mixing-induced error terms and culminate in a full-measure conclusion. Minor clarifications would further improve readability and distinguish the result from stronger (not claimed) strong Borel–Cantelli laws.