  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.
