On a problem of W. J. LeVeque concerning metric diophantine approximation

Abstract:

We consider the diophantine approximation problem \[ \left \vert x-\frac {p}{q}\right \vert \leq \frac {f(\log q)}{q^2} \] where $f$ is a fixed function satisfying suitable assumptions. Suppose that $x$ is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdős). Here, we are going to extend and solve LeVeque’s problem.