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 ${\dim }_B\left(F\times G\right)\le {\dim }_B\left(F\right)+{\dim }_B\left(G\right)$ whilst the lower box-counting dimension satisfies the inequality ${\dim }_{LB}\left(F\times G\right)\ge {\dim }_{LB}\left(F\right)+{\dim }_{LB}\left(G\right)$. We construct Cantor-like sets to demonstrate that both of these inequalities can be strict.