Remarks on dimensions of Cartesian product sets

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.