The Dimensions of Hyperspaces

We use the theory of computing to prove general hyperspace dimension theorems for three important fractal dimensions. Let $X$ be a separable metric space, and let $\mathcal{K}(X)$ be the \emph{hyperspace} of $X$, i.e., the set of all nonempty compact subsets of $X$, endowed with the Hausdorff metric. For the lower and upper Minkowski (i.e., box-counting) dimensions, we give precise formulas for the dimension of $\mathcal{K}(E)$, where $E$ is any subset of $X$. For packing dimension, we give a tight bound on the dimension of $\mathcal{K}(E)$ in terms of the dimension of $E$, where $E$ is any analytic (i.e., $\mathbf{\Sigma}^1_1$) subset of $X$. These results also hold for a large class of gauge families (Hausdorff families of gauge functions). The logical structures of our proofs are of particular interest. We first extend two algorithmic fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to individual points $x \in X$--to arbitrary separable metric spaces and to arbitrary gauge families. We then extend the point-to-set principle of J. Lutz and N. Lutz (2018) to arbitrary separable metric spaces and to a large class of gauge families. Finally, we use this principle to prove our hyperspace packing dimension theorem. This is one of a handful of cases--all very recent and all using the point-to-set principle--in which the theory of computing has been used to prove new theorems in classical geometric measure theory, theorems whose statements do not involve computation or logic.