  We study the dimension properties of the spectral measure of the Circular
$\beta$-Ensembles. For $\beta \geq 2$ it it was previously shown by Simon that
the spectral measure is almost surely singular continuous with respect to
Lebesgue measure on $\partial \mathbb{D}$ and the dimension of its support is
$1 - 2/\beta$. We reprove this result with a combination of probabilistic
techniques and the so-called Jitomirskaya-Last inequalities. Our method is
simpler in nature and mostly self-contained, with an emphasis on the
probabilistic aspects rather than the analytic. We also extend the method to
prove a large deviations principle for norms involved in the Jitomirskaya-Last
analysis.
