For processes $X(t)$ with stable components we calculate $\dim X(E)$ in terms of $\dim E$, where $E$ is a fixed Borel subset of [0, 1] of known Hausdorff-Besicovitch dimension, $\dim E$. Our results extend the earlier ones of Blumenthal and Getoor in the stable case.
"A Dimension Theorem for Sample Functions of Processes with Stable Components." Ann. Probab. 1 (5) 849 - 853, October, 1973. https://doi.org/10.1214/aop/1176996850