Systems of Rank One, Explicit Rokhlin Towers, and Covering Numbers

Rotations $f_\alpha$ of the one-dimensional torus (equipped with the normalized Lebesgue measure) by an irrational angle $\alpha$ are known to be dynamical systems of rank one. This is equivalent to the property that the covering number $F^*(f_\alpha)$ of the dynamical system is one. In other words, there exists a basis $B$ such that for arbitrarily high $h$ an arbitrarily large proportion of the unit torus can be covered by the Rokhlin tower $(f_\alpha^kB)_{k=0}^{h-1}$. Although $B$ can be chosen with diameter smaller than any fixed $\varepsilon > 0$, it is not always possible to take an interval for $B$ but this can only be done when the partial quotients of $\alpha$ are unbounded. In the present paper, we ask what maximum proportion of the torus can be covered when $B$ is the union of $n_B \in \mathbb{N}$ disjoint intervals. This question has been answered in the case $n_B =1$ by Checkhova, and here we address the general situation. If $n_B = 2$ we give a precise formula for the maximum proportion. Furthermore, we show that for fixed $\alpha$ the maximum proportion converges to $1$ when $n_B \to \infty$. Explicit lower bounds can be given if $\alpha$ has constant partial quotients. Our approach is inspired by the construction involved in the proof of the Rokhlin Lemma and furthermore makes use of the Three Gap Theorem.