Deviations of ergodic sums for toral translations II. Boxes

Abstract

We study the Kronecker sequence $\{n\alpha \}_{n\leq N}$ on the torus ${\mathbf {T}}^{d}$ when $\alpha $ is uniformly distributed on ${\mathbf {T}}^{d}$ . 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.