Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer

TL;DR

This paper develops a new version of the Borel-Cantelli lemma applicable to arbitrary probability spaces, proving zero-one laws and applying these results to solve inhomogeneous Duffin-Schaeffer conjecture cases in number theory and dynamical systems.

Contribution

It introduces a generalized zero-one law under mild conditions, extending applicability to arbitrary probability spaces and resolving key conjectures in number theory.

Findings

Established a new zero-one law for inhomogeneous events.

Resolved the weak inhomogeneous Duffin-Schaeffer conjecture.

Provided new characterizations of Borel-Cantelli sequences in dynamical systems.

Abstract

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.