Remarks on dimensions of Cartesian product sets

TL;DR

This paper explores how the dimensions of Cartesian product sets can vary within known bounds, providing examples that attain all possible values permitted by established inequalities.

Contribution

It constructs specific examples of compact sets demonstrating that the dimension of their product can reach any value within the theoretical bounds.

Findings

Examples show the dimension of product sets can attain all values within bounds.

The study uses digit restriction sets to analyze dimension behavior.

Results clarify the range of possible dimensions for Cartesian products.

Abstract

Given metric spaces $E$ and $F$, it is well known that $$\dim_HE+\dim_HF\leq\dim_H(E\times F)\leq\dim_HE+\dim_PF,$$ $$\dim_HE+\dim_PF\leq \dim_P(E\times F)\leq\dim_PE+\dim_PF,$$ and $$\underline{\dim}_BE+\overline{\dim}_BF \leq\overline{\dim}_B(E\times F) \leq\overline{\dim}_BE+\overline{\dim}_BF,$$ where $\dim_HE$, $\dim_PE$, $\underline{\dim}_BE$, $\overline{\dim}_BE$ denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of $E$, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product $E\times F$ may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.