= Cartesian powers of the Cantor set
{c}
{title2=$\dim_HC^d=d\log2/\log3$}
For the middle-third <Cantor set> $C$ and positive integer $d$, $C^d$ is compact and <totally disconnected>. A connected subset projects to a point in each coordinate and is therefore itself a point. At level $n$, the product has $2^{dn}$ cubes of side $3^{-n}$. These covers give dimension at most $d\log2/\log3$. The product digit probability assigns each cube mass $2^{-dn}$; a ball of radius comparable to $3^{-n}$ meets only a bounded number of cubes, so its mass is at most a constant times $r^{d\log2/\log3}$. The cover-cost argument for <Hausdorff dimension> proves the matching lower bound. Thus compact totally disconnected Euclidean subsets can have arbitrarily large Hausdorff dimension when the ambient dimension is allowed to increase.
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