Strict inequality in the box-counting dimension product formulas

TL;DR

This paper constructs specific Cantor-like sets to demonstrate that the known inequalities for the box-counting dimensions of Cartesian products can be strict, showing the inequalities are not always equalities.

Contribution

It provides explicit examples of sets where the inequalities for box-counting dimensions of products are strict, clarifying the limitations of existing dimension formulas.

Findings

Examples of sets with strict inequality in box-counting dimension formulas

Demonstration that inequalities can be strict in both upper and lower dimensions

Clarification of the bounds in box-counting dimension product formulas

Abstract

It is known that the upper box-counting dimension of a Cartesian product satisfies the inequality $\dim_{B}\left(F\times G\right)\leq \dim_{B}\left(F\right) + \dim_{B}\left(G\right)$ whilst the lower box-counting dimension satisfies the inequality $\dim_{LB}\left(F\times G\right)\geq \dim_{LB}\left(F\right) + \dim_{LB}\left(G\right)$. We construct Cantor-like sets to demonstrate that both of these inequalities can be strict.