arXiv:2205.12366v2
Dynamical Borel-Cantelli Lemma for Recurrence under Lipschitz Twists
math.DSmath.NT37B2037D2037E05
Abstract
In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The zero-positive-measure dynamical Borel–Cantelli law, its full-measure upgrades under commutation or pseudo-Markov hypotheses, and the beta-transformation application are correct under the paper's regularity and mixing assumptions.
Theorems 1.3 and 1.4Correct
The convergence-divergence dichotomy
Pages 4–5 · Theorems 1.3–1.4 · arXiv:2205.12366v2
Summability gives measure zero by the first Borel–Cantelli lemma. In the divergent case, decay of correlations bounds the second moment by the square of the expectation plus a linear error, giving positive measure through the second-moment lemma. When the target map commutes with the dynamics, the limsup set is invariant modulo null sets, and ergodicity upgrades positive measure to full measure.
Full paper, version 2 ↗Theorems 1.5 and 1.6Correct
Pseudo-Markov and beta-transformation full-measure laws
Pages 5–6 and Sections 5–6 · Theorems 1.5–1.6 · arXiv:2205.12366v2
The pseudo-Markov covering property transfers positive density from a ball to the ambient space, so the limsup set has full measure. For beta transformations the cylinder estimates and invariant-density bounds verify the hypotheses uniformly, yielding the claimed zero-one law for the stated shrinking targets.
02Proofs2 reported findingsCorrect
The first- and second-moment estimates, density argument, and beta-cylinder verification are correct and complete.
Sections 2–4Correct and complete
Correlation estimates close the divergent case
Sections 2–4 · arXiv:2205.12366v2
The short-return and long-return pairs are separated at a scale for which the geometric regularity estimate controls the former and decay of correlations controls the latter. Their combined contribution is of the required order, so the Paley–Zygmund-type lower bound is uniform along a divergent subsequence.
Sections 5–6Correct and complete
Positive measure is upgraded to full measure
Sections 5–6 · arXiv:2205.12366v2
The pseudo-Markov atoms provide controlled images and bounded distortion, allowing the density theorem to contradict a complement of positive measure. In the beta case, full cylinders occur at the needed scales and the invariant measure is uniformly comparable to Lebesgue measure, verifying every hypothesis used in that argument.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.