Article
Deviations of ergodic sums for toral translations II. Boxes
Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 293-352

We study the Kronecker sequence {nα} n≤N on the torus 𝐓 d when α is uniformly distributed on 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 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.

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DOI : 10.1007/s10240-020-00120-2 
@article{PMIHES_2020__132__293_0,
     author = {Dolgopyat, Dmitry and Fayad, Bassam},
     title = {Deviations of ergodic sums for toral translations {II.} {Boxes}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {293--352},
     year = {2020},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {132},
     doi = {10.1007/s10240-020-00120-2},
     zbl = {1473.37012},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-020-00120-2/}
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Dolgopyat, Dmitry; Fayad, Bassam. Deviations of ergodic sums for toral translations II. Boxes. Publications Mathématiques de l'IHÉS, Tome 132 (2020), pp. 293-352. doi: 10.1007/s10240-020-00120-2

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