This principle has been used to obtain strengthened lower bounds on the Hausdorff dimensions of generalized Furstenberg sets, extend the fractal intersection formula for Hausdorff dimension from Borel sets to arbitrary sets, and prove that Marstrand’s projection theorem for Hausdorff dimension holds for any set E whose Hausdorff and packing dimensions coincide, whether or not E is analytic. The point-to-set principle, described below, has enabled several recent developments that do exactly this. It is rare for the theory of computing to be used to answer open mathematical questions whose statements do not involve computation or related aspects of logic. Measure theory, theorems whose statements do not involve computation or logic. Theory of computing has been used to prove new theorems in classical geometric This is one of a handful ofĬases–all very recent and all using the point-to-set principle–in which the Our hyperspace packing dimension theorem. Lutz (2018) to arbitrary separable metric spacesĪnd to a large class of gauge families. We firstĮxtend two algorithmic fractal dimensions–computability-theoretic versions ofĬlassical Hausdorff and packing dimensions that assign dimensions (x) andĭim(x) to individual points x ∈ X–to arbitrary separable metric The logical structures of our proofs are of particular interest. Large class of gauge families (Hausdorff families of gauge functions). 𝒦(E) in terms of the dimension of E, where E is any analytic Lower and upper Minkowski (i.e., box-counting) dimensions, we give preciseįormulas for the dimension of 𝒦(E), where E is any subset of X.įor packing dimension, we give a tight bound on the dimension of Nonempty compact subsets of X, endowed with the Hausdorff metric. Let X be a separable metric space,Īnd let 𝒦(X) be the hyperspace of X, i.e., the set of all Math., to appear.We use the theory of computing to prove general hyperspace dimension theoremsįor three important fractal dimensions.
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