The point-to-set principle has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces RnRn. These are classical questions, meaning that their statements do not involve computation or related aspects of logic.
In this talk I will describe the extension of two algorithmic fractal dimensions --- computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x)dimf0(x) and Dim(x)Dimf0(x) to individual points x∈Xx∈X --- to arbitrary separable metric spaces and to arbitrary gauge families. I will then discuss the extension of the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. Finally, I will indicate how the extended point-to-set principle can be used to prove new theorems about classical fractal dimensions in hyperspaces.
This is joint work with Neil Lutz and Elvira Mayordomo.