Remarks on dimensions of Cartesian product sets

Abstract

Given metric spaces $E$ and $F$, it is well known that $${\dim }_{H}E+{\dim }_{H}F\le {\dim }_H(E\times F)\le {\dim }_{H}E+{\dim }_{P}F,$$ $${\dim }_{H}E+{\dim }_{P}F\le {\dim }_P(E\times F)\le {\dim }_{P}E+{\dim }_{P}F,$$ and $${\underline{\dim }}_{B}E+{\overline{\dim }}_{B}F\le {\overline{\dim }}_B(E\times F)\le {\overline{\dim }}_{B}E+{\overline{\dim }}_{B}F,$$ where ${\dim }_{H}E$, ${\dim }_{P}E$, ${\underline{\dim }}_{B}E$, ${\overline{\dim }}_{B}E$ 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.

13 pages

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