The Dimensions of Hyperspaces
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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 𝒦(X) be the 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 𝒦(E), where E is any subset of X. For packing dimension, we give a tight bound on the dimension of 𝒦(E) in terms of the dimension of E, where E is any analytic (i.e., Σ^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 (x) and Dim(x) to individual points x ∈ 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.
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