arXiv:1904.08584v4
Zero-one laws for eventually always hitting points in in rapidly mixing systems
math.DS
Abstract
In this work we study the set of eventually always hitting points in shrinking target systems. These are points whose long orbit segments eventually hit the corresponding shrinking targets for all future times. We focus our attention on systems where translates of targets exhibit near perfect mutual independence, such as Bernoulli schemes and the Gauss map. For such systems, we present tight conditions on the shrinking rate of the targets so that the set of eventually always hitting points is a null set (or co-null set respectively).
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The zero--one criterion for eventually-always-hitting sets under the stated long-term independence hypothesis, and its product, Bernoulli, and Gauss-map applications, are correct.
Theorem 1.1Correct
The general eventually-always-hitting criterion is valid
Page 4 and Section 3 (pages 11--18) · Theorem 1.1 and its proof · arXiv:1904.08584v4
Writing for the event that the first iterates miss , the complement of the eventually-always-hitting set is . The long-term independence estimate permits a blocking argument with asymptotically multiplicative conditional probabilities. The convergent hypothesis makes the selected miss events summable, while the divergent hypothesis supplies sufficiently separated miss events with a divergent sum and controlled correlations. The ergodic zero--one law then upgrades the resulting positive-measure conclusion to the asserted zero or full measure.
Theorems 1.2--1.6Correct
The model systems satisfy the announced criteria
Pages 4--7 and Sections 4--6 (pages 18--27) · product, Bernoulli, and Gauss-map applications · arXiv:1904.08584v4
For product targets, independence gives exactly. For Bernoulli cylinders the argument separates blocks by the cylinder length, producing the exponent used in the criterion. For Gauss targets, exponential mixing after a gap of the chosen length supplies the required relative error, and gives the displayed thresholds. These computations match the hypotheses and conclusions of the general theorem.
02Proofs2 reported findingsCorrect
The blocking, conditional-measure, correlation, and model-specific mixing arguments are correct and complete; omitted elementary estimates are recoverable from the displayed bounds.
Sections 2--3Correct and complete
The zero--one law and blocking construction close both directions
Pages 8--18 · general properties and proof of the main technical result · arXiv:1904.08584v4
Nested targets make the relevant miss events monotone in the needed variables. The proof selects logarithmically separated blocks so that the gap is negligible relative to the usable orbit segment, applies the relative mixing estimate to the generated algebras, and controls the accumulated error by . The convergence and divergence estimates are applied in their correct directions, with no unhandled endpoint or dependence case.
Sections 4--6Correct and complete
The independence and rapid-mixing inputs are used within their stated ranges
Pages 18--27 · applications to independent targets, Bernoulli schemes, and the Gauss map · arXiv:1904.08584v4
The product and Bernoulli applications compute miss probabilities on genuinely disjoint coordinate blocks. In the Gauss-map application, the chosen gap simultaneously satisfies the quantitative mixing requirement and the upper bound required in the general theorem; the remaining finite initial indices do not affect any limsup set or series test.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.