Hausdorff measure of cartesian product of Cantor sets

TL;DR

The paper studies bounds on the $s_d$-dimensional Hausdorff measure of the $d$-fold Cartesian product of the ternary Cantor set, $\mathcal{C}^d$, where $s_d = d\log_3(2)$. It leverages self-similarity under the strong open set condition and Moran-type results to derive both refined upper bounds (numerically computed for $d=2$ to $8$) and a novel lower-bound framework based on repulsive pairs within the iterated function system, enabling explicit lower bounds in low dimensions. The authors show, for instance, that $\mathcal{H}^{s_3}(\mathcal{C}^3) > 1.811621$, surpassing $\mathcal{H}^{s_2}(\mathcal{C}^2) \approx 1.48329$, demonstrating that higher-dimensional Cantor products can have strictly larger higher-dimensional Hausdorff measures than lower-dimensional ones. Overall, the work extends the understanding of fractal measures of Cantor-product sets by providing concrete upper and lower bounds for small $d$, and introduces a practical methodology for tightening these bounds via optimal-set and repulsive-pair analyses.

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 $d$.

Figures (5)

• Figure 1: The Ternary Cantor set $\mathcal{C}$, drawn to stage 3.
• Figure 2: Graph of $\mathcal{H}^s(F)$ against $s$ with jump at its dimension
• Figure 3: Labeling of the Quadrants in level 1 basic sets of $\mathcal{C}^3$.
• Figure 4: Naïve Upper Bound for the Hausdorff Measure for $\mathcal{C}^d$ with $d = 1,\cdots, 6$.
• Figure 5: Computed Improvements on the Upper Bound (red) compared to the Naïve Upper Bound (blue).

Theorems & Definitions (32)

• Definition 1.1: Ternary Cantor Set
• Definition 1.2: $s$-dimensional Hausdorff Measure
• Proposition 1.3: Section 2.2 in Falconer
• Definition 1.4: Hausdorff Dimension
• Definition 1.5
• Definition 1.6: Open Set Condition
• Theorem 1.7
• Definition 1.8: Strong Open Set Condition
• Theorem 1.9
• Definition 1.10: Optimal Set