I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture (http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf, http://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture). I have found only one paper that seems to follow this thread, namely the PHD Thesis of Alan Kaan Haynes, found here: http://www.lib.utexas.edu/etd/d/2006/haynesa79646/haynesa79646.pdf
In the specific case of the Duffin-Schaeffer conjecture, the key to the problem is to show that the sets involved are 'nearly' independent, so one can apply the other direction of the Borel-Cantelli Lemma to get the desired result. Because independence is inherently a probabilistic idea, the exploration of probabilistic tools in application to metric number theory might be fruitful. The Haynes thesis explores use of martingales, for example.
So I am wondering if there are other papers or books that explores use of probabilistic tools in metric number theory?