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The Hausdorff dimension of non-uniquely ergodic directions in<i>H</i>(2) is almost everywhere 1<i>โ</i>2
We show that for almost every (with respect to Masur-Veech measure) $\omega \in \mathcal{H}(2)$, the set of angles $\theta \in [0, 2\pi)$ so that $e^{i\theta}\omega$ has non-uniquely ergodic vertical foliation has Hausdorff dimension (and codimension) $1/2$.