Variants of the Second Borel-Cantelli Lemma and their Applications in Metric Number Theory

In Metric Number Theory we are concerned with the arithmetical properties of almost all numbers (in the first case with respect to Lebesgue measure on the real line, but the ideas generalise to ℝk and other situations). Investigations therefore involve both analysis and number theory. It is the purpose of this paper to review the important contribution made by variants of the second Borel-Cantelli Lemma (also known as the divergence part of the Borel-Cantelli Lemma). While doing this we shall prove sharper variants than have previously appeared and give their applications.