Limit Theorems for a Class of Inequalities Arising from the Metric Theory of Diophantine Approximation

Abstract

In a series of papers appeared in the late 50's of the last century, LeVeque investigated the probabilistic behavior of the solution set of inequalities arising from the metric theory of diophantine approximation. Amongst other results, he established a central limit theorem under an additional restriction for the solutions. Furthermore, he conjectured a similar result to be true when this additional restriction is dropped; a situation that is far more important for the metric theory of diophantine approximation. In my talk I will first summarize previous results in this area and then present the main ideas of my recent proof of the above mentioned conjecture. In particular, I will work out the main difficulties that had to be overcome. Furthermore, I will present stronger probabilistic results for the solution sets of these inequalities (such as the law of the iterated logarithm, invariance principles etc.) that improve and extend most of the results deduced by and after LeVeque.