The Hausdorff Dimension of Non-Uniquely Ergodic directions in   $\mathcal{H}(2)$ is almost everywhere $1/2$
  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) $1/2$. 
Jayadev S. Athreya and Jon Chaika
math.DS