1/2-Heavy Sequences Driven By Rotation

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 $d$.