QUANTITATIVE TWISTED RECURRENCE PROPERTIES FOR PIECEWISE EXPANDING MAPS ON [0, 1]d

Jiachang Li, Chao Ma

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper establishes zero–one laws for twisted shrinking targets (hyperrectangles and hyperboloids) for piecewise expanding maps with exponentially mixing ACIPs under the stated density hypotheses (Theorems 1.2 and 1.3), via local “untwisting,” Zygmund differentiation for rectangles, sharp hyperboloid volume asymptotics, and a second-moment/Paley–Zygmund argument to prove divergence; see the statements and proof skeleton around Theorems 1.2–1.3 and Sections 3–4 (e.g., Lemma 3.2 for localization, the Zygmund-based bounds, and the second-moment estimates) . The candidate’s solution follows the same architecture: discretize/untwist, compare μ(target) to model volumes, and apply a strong Borel–Cantelli/second-moment scheme. Differences are largely technical (e.g., citing Lebesgue rather than Zygmund for rectangles, and a slightly different mixing-error bookkeeping), but they do not change the substance or the conclusions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes sharp zero–one laws for twisted shrinking targets in a non-conformal (piecewise expanding) setting, unifying recurrence and shrinking targets and treating hyperboloids with the correct δ(−log δ)\^{d−1} scaling. The methods are standard but well-adapted: a careful untwisting discretization, Zygmund differentiation for rectangles, volume asymptotics for hyperboloids, and a robust second-moment scheme under exponential mixing. I recommend minor revisions to tidy a few presentation points (terminology consistency for mixing, a brief standalone statement of the hyperboloid volume asymptotics with constants, and a clarifying remark on the Zygmund vs. Lebesgue differentiation basis).