Deviations of ergodic sums for toral translations II. Boxes

We study the Kronecker sequence $\{n\alpha {\}}_{n\le N}$ on the torus ${\mathbb{T}}^d$ when $\alpha$ is uniformly distributed on ${\mathbb{T}}^d\mathrm{.}$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by ${\ln }^{d}N$, converges as $N\to \mathrm{\infty }$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.