  We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus
${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We
show that the discrepancy of the number of visits of this sequence to a random
box, normalized by $\ln^d N$, converges as $N\to\infty$ 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.
