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

TL;DR

This paper investigates the Hausdorff dimension and measure of Cantor-type sets generated by $eta$-expansions, demonstrating their continuity and finiteness, thus advancing understanding of fractal geometry in non-integer bases.

Contribution

It provides explicit calculations of Hausdorff dimension and measure for $eta$-expansion Cantor sets, showing their continuity with respect to the base $eta$.

Findings

Hausdorff dimension $d$ is continuous in $eta$

Hausdorff measure of the sets is finite and positive

Results extend understanding of fractal properties in non-integer bases

Abstract

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.