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

Abstract

In an earlier paper, we solved LeVeque's problem, to establish a central limit theorem for the number of solutions of the diophantine inequality \[ \left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \] in unknowns $p,q$ with $q\,{>}\,0$, where $f$ is a function satisfying special assumptions and $x$ is chosen randomly in the unit interval. Here, we are interested in the almost sure behavior of the solution set. In particular, we obtain a generalized law of the iterated logarithm and we prove a result that gives strong evidence that the law of the iterated logarithm with the standard norming sequence (suggested by the central limit theorem) holds as well. Both results have to be compared with a theorem of W. M. Schmidt; they imply an inverse to Schmidt's theorem and a strong law of large numbers with an error term that is essentially better than the one provided by Schmidt's result.