Extending the Reach of the Point-To-Set Principle

Lutz, Jack H ; Lutz, Neil ; Mayordomo Cámara, Elvira (Universidad de Zaragoza)

Resumen: 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., Σ1 1, a tight bound on the packing dimension of the hyperspace of E in terms of the packing dimension of E itself.

Idioma: Inglés

DOI: 10.4230/LIPIcs.STACS.2022.48

Año: 2022

Publicado en: Leibniz international proceedings in informatics 219 (2022), 48 [14 pp.]
ISSN: 1868-8969

Factor impacto SCIMAGO: 0.832 - Software - Logic

Financiación: info:eu-repo/grantAgreement/ES/DGA/T64-20R
Financiación: info:eu-repo/grantAgreement/ES/MICINN AEI/PID2019-104358RBI00
Financiación: info:eu-repo/grantAgreement/ES/MICINN AEI/TIN2016-80347-R

Tipo y forma: Comunicación congreso (Versión definitiva)

Área (Departamento): Área Lenguajes y Sistemas Inf. (Dpto. Informát.Ingenie.Sistms.)