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.

中文翻译：

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口头翻译的各态历经和的偏差 II。盒子

我们研究环面\({\mathbf {T}}^{d}\ ) 上的克罗内克序列\(\{n\alpha \}_{n\leq N}\)，当\(\alpha \)均匀分布时分布在\({\mathbf {T}}^{d}\)上。我们表明，该序列对随机框的访问次数的差异，通过\(\ln ^{d} N\)归一化，收敛为\(N\to \infty \)到柯西分布。证明的关键要素是\(d+1\)维晶格空间上的嘉当作用的泊松极限定理。