Hausdorff measure of cartesian product of Cantor sets

Abstract

Hausdorff measure and Hausdorff dimension are useful tools to describe fractals. This paper investigates the bounds on the $d{\log }_{3}2$-dimensional Hausdorff measure of the $d$-fold Cartesian product of the $1/3$ Cantor set, ${\mathcal{C}}^d$. By applying known theorems on the Hausdorff measure of fractals satisfying the strong open set condition and generalizing what has been done on ${\mathcal{C}}^2$, we compute stricter upper and lower bounds for the Hausdorff measure of ${\mathcal{C}}^d$ for several small integers $d$.

13 pages, 5 figures