Borel-Cantelli sequences

Abstract

A sequence {x n } 1 ∞ in the unit interval [0, 1) = ℝ/ℤ is called Borel-Cantelli, or BC, if for all non-increasing sequences of positive real numbers {a n } with 48001013 1. $$\sum\nolimits_{i = 1}^\infty {} {a_i} = \infty $$ , the set $$\{ x \in [0,1)\left| {\left| {x - {x_n}} \right|} \right. < {a_n}{\rm{for infinitely many }}n \ge 1\} $$ has full Lebesgue measure. (Speaking informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds). The notion of BC sequences is motivated by the monotone shrinking target property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC sequences and sequences that are not BC are also presented. The property of a sequence to be BC is a delicate Diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC, while the orbits of a “generic” IET are not. The notion of BC sequences is extended from [0, 1) to sequences in Ahlfors regular spaces.