Deviations of Ergodic Sums for Toral Translations Ii . Boxes

@inproceedings{Dolgopyat2012DeviationsOE,
  title={Deviations of Ergodic Sums for Toral Translations Ii . Boxes},
  author={Dmitry Dolgopyat and BASSAM FAYAD},
  year={2012}
}
We study the Kronecker sequence {nα}n≤N on the torus T when α is uniformly distributed on T. We show that the discrepancy of the number of visits of this sequence to a random box, normalized by lnN , converges as N → ∞ 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.