Limiting distribution of visits of sereval rotations to shrinking intervals

TL;DR

This paper investigates the asymptotic distribution of visits by multiple rotations to shrinking intervals on the unit circle, revealing independence properties and generalizing previous results by Marklof.

Contribution

It extends earlier work by Marklof to multiple rotations and provides a joint limiting distribution for visits to shrinking intervals.

Findings

Visit counts follow a joint limiting distribution as trajectory lengths grow.

As the number of rotations increases, visit counts become asymptotically independent unless an arithmetic obstruction exists.

The results generalize previous findings by Marklof to multiple rotations and intervals.

Abstract

We show that given $n$ normalized intervals on the unit circle, the numbers of visits of $d$ random rotations to these intervals have a joint limiting distribution as lengths of trajectories tend to infinity. If $d$ then tends to infinity, then the numbers of points in different intervals become asymptotically independent unless an arithmetic obstruction arises. This is a generalization of earlier results of J. Marklof.