Strict Inequality in the Box-Counting Dimension Product Formulas

Abstract

We supplement the well known upper and lower box-counting product inequalities to give the new product formula \begin{align*} \dim_{LB}F + \dim_{LB}G &\leq \dim_{LB}\left(F\times G\right) \\ &\leq \min\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ &\leq \max\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ & \leq \dim_{B}\left(F\times G\right)\\ &\leq \dim_{B}F+\dim_{B}G \end{align*} for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products.