2024/06/27 by Victor Beresnevich, Beresnevich, Victor, Manuel Hauke +3 · 4 citations
Economics, Econometrics and Finance · #11A55 #11J20 #11J54 #11J70 #11J71 #11J83 #11K06 #11K38 #11K50 #11K60 #37A44 #60F20 #Dynamical Systems (math.DS) #Economic theories and models #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2406.19198
openalex publication_date 2024/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The most versatile version of the classical divergence Borel-Cantelli lemma shows that for any divergent sequence of events En in a probability space satisfying a quasi-independence condition, its corresponding limsup set E_∞ has positive probability. In particular, it provides a lower bound on the probability of E_∞. In this paper we establish a new version of this classical result which guarantees, under an additional mild assumption, that the probability of E_∞ 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.