1/2-Heavy Sequences Driven By Rotation

TL;DR

This paper studies a special set of points on the circle with a property related to their orbit under irrational rotation, revealing its Hausdorff dimension varies with the rotation angle and establishing its complexity for almost all angles.

Contribution

It introduces a renormalization approach to analyze the Hausdorff dimension of the set of points with balanced orbit distributions under irrational rotation.

Findings

Hausdorff dimension is constant for almost every rotation angle

Dimension varies densely across all values in [0,1] for certain angles

The set's dimension is strictly between zero and one for almost all rotations

Abstract

We investigate the set of $x \in S^1$ such that for every positive integer $N$, the first $N$ points in the orbit of $x$ under rotation by irrational $\theta$ contain at least as many values in the interval $[0,1/2]$ as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every $\theta$, and that for every $d \in [0,1]$ there is a dense set of $\theta$ for which the Hausdorff dimension of this set is $d$.