Published paper
Measure theoretic laws for limsup sets defined by rectangles
MSC 2020
11J8311K60
Abstract
The paper develops a general Lebesgue-measure principle for limsup sets defined by rectangles under a rectangular ubiquity hypothesis.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statements:
- Theorem 2.7 · printed pp. 5–6; proof §6, pp. 15–21The rectangular zero-full Lebesgue-measure law from which the focal paper's side theorem follows. Its divergence half runs through Lemmas 6.1–6.2 and Theorem 2.5; its convergence half is elementary Borel–Cantelli.
- Theorem 2.5 · printed p. 5; proof §3, pp. 7–12Turns rectangular ubiquity, regularity, and divergence of the volume sum into full measure. Theorem 2.7 applies it after Lemma 6.2 constructs the needed ubiquitous system.
- Lemma 6.1 · printed pp. 16–18Regularizes the approximating functions without changing the measure of the limsup set and supplies the decay condition needed for Theorem 2.5.
- Lemma 6.2 · printed pp. 19–20Constructs the ubiquitous rectangular system used in the proof of Theorem 2.7; Minkowski's theorem produces its initial nonzero integer vector.
Proof-critical source
Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
This paper is included only for the following marked statements:
- Theorem 2.7 · printed pp. 5–6; proof §6, printed pp. 15–21General Khintchine–Groshev zero–full law for rectangular approximation used as Theorem B by the focal paper.
- Theorem 2.5 · printed p. 5; proof §3, printed pp. 7–12Abstract rectangular ubiquity theorem used in the divergence proof of Theorem 2.7.
- Lemma 6.2 · printed pp. 19–20Minkowski-based ubiquity input for the rectangular Khintchine–Groshev theorem.
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