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.

💡 Research Summary

The paper investigates a broad class of Cantor‑type sets generated by deleting digits in non‑integer β‑expansions. For a fixed base β > 1, the authors consider a finite set A of admissible digits (a subset of {0,1,…,⌊β⌋}) and define the set C_{β,A} as all numbers in