  Let $q = e^{i \theta} \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and
let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that
$vu = quv$. In this paper we find the optimal constant $c = c_\theta$ such that
$u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are
commuting unitaries. We show that \[ c_\theta =
\frac{4}{\|u_\theta+u_\theta^*+v_\theta+v_\theta^*\|}, \] where $u_\theta,
v_\theta$ are the universal $q$-commuting pair of unitaries, and we give
numerical estimates for the above quantity. In the course of our proof, we also
consider dilating $q$-commuting unitaries to scalar multiples of $q'$-commuting
unitaries. The techniques that we develop allow us to give new and simple
"dilation theoretic" proofs of well known results regarding the continuity of
the field of rotations algebras. In particular, for the so-called "Almost
Mathieu Operator" $h_\theta = u_\theta+u_\theta^*+v_\theta+v_\theta^*$, we
recover the fact that the norm $\|h_\theta\|$ is a Lipshitz continuous function
of $\theta$, as well as the result that the spectrum $\sigma(h_\theta)$ is a
$\frac{1}{2}$-H\"older continuous function in $\theta$ with respect to the
Hausdorff metric. In fact, we obtain this H\"older continuity of the spectrum
for every selfadjoint $*$-polynomial $p(u_\theta,v_\theta)$, which in turn
endows the rotation algebras with the natural structure of a continuous field
of C*-algebras.
