Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer
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2024
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| author | Beresnevich, Victor Hauke, Manuel Velani, Sanju |
| author_facet | Beresnevich, Victor Hauke, Manuel Velani, Sanju |
| contents | The most versatile version of the classical divergence Borel-Cantelli lemma shows that for any divergent sequence of events $E_n$ in a probability space satisfying a quasi-independence condition, its corresponding limsup set $E_\infty$ has positive probability. In particular, it provides a lower bound on the probability of $E_\infty$. In this paper we establish a new version of this classical result which guarantees, under an additional mild assumption, that the probability of $E_\infty$ is not just positive but is one. Unlike existing optimal results, it is applicable within the setting of arbitrary probability spaces. We then go onto to consider a range of applications in number theory and dynamical systems. These include new results on the inhomogeneous Duffin-Schaeffer conjecture. In particular, we establish alternatives to the classical (homogeneous) zero-one laws of Cassels and Gallagher and use them to resolve the so-called weak Duffin-Schaeffer conjecture for an arbitrary rational inhomogeneous shift. As a bi-product, we establish the Duffin-Schaeffer conjecture with congruence relations. The applications to dynamical systems include new characterisations of Borel-Cantelli sequences and new dynamical Borel-Cantelli lemmas, as well as characterising Khintchine-type sequences for shrinking targets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer Beresnevich, Victor Hauke, Manuel Velani, Sanju Number Theory Dynamical Systems Probability 11J83, 11K60, 11J20, 60F20, 37A44, 11J71, 11A55, 11J70, 11K38, 11J54, 11K06, 11K50 The most versatile version of the classical divergence Borel-Cantelli lemma shows that for any divergent sequence of events $E_n$ in a probability space satisfying a quasi-independence condition, its corresponding limsup set $E_\infty$ has positive probability. In particular, it provides a lower bound on the probability of $E_\infty$. In this paper we establish a new version of this classical result which guarantees, under an additional mild assumption, that the probability of $E_\infty$ is not just positive but is one. Unlike existing optimal results, it is applicable within the setting of arbitrary probability spaces. We then go onto to consider a range of applications in number theory and dynamical systems. These include new results on the inhomogeneous Duffin-Schaeffer conjecture. In particular, we establish alternatives to the classical (homogeneous) zero-one laws of Cassels and Gallagher and use them to resolve the so-called weak Duffin-Schaeffer conjecture for an arbitrary rational inhomogeneous shift. As a bi-product, we establish the Duffin-Schaeffer conjecture with congruence relations. The applications to dynamical systems include new characterisations of Borel-Cantelli sequences and new dynamical Borel-Cantelli lemmas, as well as characterising Khintchine-type sequences for shrinking targets. |
| title | Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer |
| topic | Number Theory Dynamical Systems Probability 11J83, 11K60, 11J20, 60F20, 37A44, 11J71, 11A55, 11J70, 11K38, 11J54, 11K06, 11K50 |
| url | https://arxiv.org/abs/2406.19198 |