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Contemporary Problems in Number Theory
October 4, 2012 12:45, Moscow, Steklov Mathematical Institute, Room 530 (8 Gubkina)

Limit distributions of normalized discrepancies of Kronecker sequences

Bassam Fayad

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Abstract: We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus $\mathbb T^d$ where $\alpha$ is uniformly distributed on $\mathbb T^d$. In the one dimensional case, Kesten proved in the 1960s that the discrepancies of the number of visits of the Kronecker sequence to a random interval, normalized by $\ln N$, converges to a Cauchy distribution.
There are two possible generalizations of the problem to higher dimensions : by either considering visits to balls, or to cubes. The behavior of the discrepancy is related to the so called small denominators of the corresponding frequency, and the difficulty in higher dimensions arises from the absence of a continued fraction theory with powerful metrical results as in the one dimension case.
We rather study both situations, balls and cubes, using the ergodic theory of flows on homogeneous spaces. In the case of balls, we show that the discrepancy normalized by $N^{(d-1)/2d}$ converges in distribution to a nonstandard law. In the case of cubes, we show that the discrepancy normalized by $(\ln N)^d$ converges to a Cauchy distribution. The key ingredient of the latter proof is a Poisson limit theorem for the Cartan action on the space of $(d+1)$-dimensional lattices.

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