Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

14
The categories of this paper are: math-ph
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs