  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.
