Borel-Cantelli sequences

A sequence $\{x_n{\}}_1^{\mathrm{\infty }}$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $a_i=\mathrm{\infty }$ the set $$B(x_n,a_n))=\{x\in [0,1)\mid \mathrm{\mid }{x}_n-x\mathrm{\mid }<a_n\text{for}\mathrm{\infty }\text{many}n\ge 1\}$$ has full Lebesgue measure. (To put it 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 and not BC sequences are 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 to more general spaces.