Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets

We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\beta$. The $d$-dimentional Hausdorff measure of these sets is finite and positive.