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

We consider the diophantine approximation problem

where is a fixed function satisfying suitable assumptions. Suppose that 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 Erdos). Here, we are going to extend and solve LeVeque's problem.