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.
Categories: math.DS math.NT