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Dimension Results for the Spectral Measure of the Circular Beta Ensembles

We study the dimension properties of the spectral measure of the Circular $β$-Ensembles. For $β\geq 2$ 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 $1 - 2/β$. 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.