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 $d\log_32$-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 $d$.