A Limit Theorem for Birkoff sums of non-integrable functions over rotations

We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N=S_N(\alpha ,x)$ be the N^th non-renormalized Birkhoff sum, where $xin[0,1)$ is the initial point, $\alpha \in [0,1)$ is the rotation number and $(\alpha ,x)$ are uniformly distributed. We prove that $S_N\mathrm{/}N$ has a joint limiting distribution in $(\alpha ,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.