We establish an upper bound on the asymptotic probability of an $SLE(\kappa)$ curve hitting two small intervals on the real line as the interval width goes to zero, for the range $4 < \kappa < 8$. As a consequence we are able to prove that the random set of points in $R$ hit by the curve has Hausdorff dimension $2-8/\kappa$, almost surely.
"Hausdorff Dimension of the SLE Curve Intersected with the Real Line." Electron. J. Probab. 13 1166 - 1188, 2008. https://doi.org/10.1214/EJP.v13-515