  Given an $\alpha > 1$ and a $\theta$ with unbounded continued fraction
entries, we characterise new relations between Sturmian subshifts with slope
$\theta$ with respect to (i) an $\alpha$-H\"oder regularity condition of a
spectral metric, (ii) level sets defined in terms of the Diophantine properties
of $\theta$, and (iii) complexity notions which we call $\alpha$-repetitive,
$\alpha$-repulsive and $\alpha$-finite; generalisations of the properties known
as linearly repetitive, repulsive and power free, respectively. We show that
the level sets relate naturally to (exact) Jarn\'{\i}k sets and prove that
their Hausdorff dimension is $2/(\alpha + 1)$.
