Extending the Reach of the Point-to-Set Principle
نویسندگان
چکیده
The point-to-set principle of J. Lutz and N. (2018) has recently enabled the theory computing to be used answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that their statements do not involve computation or related aspects logic. In this paper we extend reach from arbitrary separable metric $X$. We first two dimensions--computability-theoretic versions Hausdorff packing dimensions assign $\dim(x)$ $\textrm{Dim}(x)$ individual points $x\in X$--to gauge families. Our main results then a large class We demonstrate power our extended by using it prove new theorems hyperspaces. (For concrete computational example, stages $E_0, E_1, E_2, \ldots$ construct self-similar $E$ plane elements hyperspace plane, they converge hyperspace.) third result, proven via principle, states that, under wide variety families, dimension agrees with upper Minkowski on all hyperspaces compact sets. use theorem give, for sets analytic, i.e., $\mathbf{\Sigma}^1_1$, tight bound terms itself.
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ژورنال
عنوان ژورنال: Social Science Research Network
سال: 2022
ISSN: ['1556-5068']
DOI: https://doi.org/10.2139/ssrn.4254623