Question

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?

Answer 1

When you say "metric number theory" I am not sure if you mean probabilistic results from the scope of all of number theory, or if you are referring specifically to problems in Diophantine approximation. In my mind the adjective "metric" is really the same as "probabilistic." Either way though, the answer to your question is definitely yes, there are lots of people who have studied number theory (and in particular Diophantine approximation) with the background and point of view of probability theory. In fact there is an entire MSC category, 11K, which is devoted to this topic.

A lot of important techniques and results about current open problems in this subject are not in any book, so the best thing if you have a particular problem in mind is to talk to people and search for papers. However if you are looking for books that give more of a general picture, there are a couple standard references:

1) "Metric number theory," Glyn Harman,

2) "Introduction to analytic and probabilistic number theory," Gerald Tenenbaum.

Final comment: The fact that martingales seem not to have made an appearance in number theory until fairly recently is a slight anomaly. As far as I am aware the first occurrence of this event is in a paper by Losert and Tichy from 1986, "On uniform distribution of subsequences," where they use martingales as well as drawing heavily on several other results and ideas from probability theory.