Is there a sort of distributional estimate in Diophantine approximation which allows to estimate the number of solutions which provide a certain quality of approximation? For example, how large is the measure of the numbers $x \in [0,1]$ for which $$ \# \Big\{ 1 \leq n \leq N: \|n x\| \leq 1/N \Big\} \geq \log N? $$ (Here, of course, the $\log N$ on the right-hand side could be any other term as well, say $(\log N)^{1/2}$ or whatever. Or the $1/N$ inside the brackets could be $1/(N \log N)$. Note that the usual methods from Diophantine approximation allow to estimate the desired measure in the case when the right-hand side is 1. By $\| \cdot \|$ we denote the distance to the nearest integer.)

If one believes the events to be somewhat independent, then the distribution we look for should follow roughly a Poisson distribution. Is something known in that direction?