Strict inequality in the 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.