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Deviations of ergodic sums for toral translations II. Boxes
Publications Mathématiques de l'IHÉS, Volume 132 (2020), pp. 293-352
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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 
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     author = {Dmitry Dolgopyat and Bassam Fayad},
     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},
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     language = {en},
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Dmitry Dolgopyat; Bassam Fayad. Deviations of ergodic sums for toral translations II. Boxes. Publications Mathématiques de l'IHÉS, Volume 132 (2020), pp. 293-352. doi: 10.1007/s10240-020-00120-2

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