# Student Probability Seminar

#### Limiting Distribution of Visits of Several Rotations to Shrinking Intervals

Let $$n$$, $$N$$, and $$d$$ be positive integers. Let $$L(m, \alpha) = \sum_{i=1}^d m_i \alpha_i (\mod{1})$$ be the trajectory of $$d$$ rotations by $$\alpha_1, . . . , \alpha_d$$ on the $$1$$-torus with $$1 \leq m_i \leq N$$. Further let $$\xi_1, . . . , \xi_n \in [0, 1)$$ and $$\sigma_1, . . . , \sigma_n > 0$$ be fixed. We interprete the numbers of visits of $$L(m, \alpha)$$ to $$(\xi_1, \xi_1 + \sigma/N), . . . ,(\xi_n, \xi_n + \sigma_n/N)$$ as a function of $$\alpha_1, . . . , \alpha_d$$, and thus a random integer vector realized on the $$d$$-cube. We prove that as $$N$$ tends to infinity, the distribution of this random vector has a limit that depends on the other parameters and has finitely many moments in each case. A subsequent limit transition can be made by letting $$d$$ tend to infinity. In this setting we prove that the new limiting distribution is the multidimensional Poisson distribution unless an arithmetic obstruction arises. There results are a generalization of earlier work of J. Marklof.