We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha $-stable Lévy processes with $1< \alpha \le 2$. This extends a theorem of Kaufman  for Brownian motion. Our method is different from that of  and depends on covering principles for Markov processes.
"Uniform Hausdorff dimension result for the inverse images of stable Lévy processes." Electron. Commun. Probab. 23 1 - 10, 2018. https://doi.org/10.1214/18-ECP180