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# Extending the Reach of the Point-To-Set Principle

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LIPIcs.STACS.2022.48.pdf
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## Acknowledgements

We thank anonymous reviewers for several observations that have improved this paper.

## Cite As

Jack H. Lutz, Neil Lutz, and Elvira Mayordomo. Extending the Reach of the Point-To-Set Principle. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 48:1-48:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.48

## Abstract

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces ℝⁿ. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces X. We first extend two fractal dimensions - computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions dim(x) and Dim(x) to individual points x ∈ X - to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages E₀, E₁, E₂, … used to construct a self-similar fractal E in the plane are elements of the hyperspace of the plane, and they converge to E in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets E that are analytic, i.e., Σ¹₁, a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computability
##### Keywords
• algorithmic dimensions
• geometric measure theory
• hyperspace
• point-to-set principle

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