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arXiv · 2510.09701

Hausdorff measure of cartesian product of Cantor sets

Siyuan Guo·Taylor Jones
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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 $d$.

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Siyuan Guo, Taylor Jones. 2025-10-09. Hausdorff measure of cartesian product of Cantor sets. https://arxiv.org/abs/2510.09701

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