000118000 001__ 118000 000118000 005__ 20230706131742.0 000118000 0247_ $$2doi$$a10.4230/LIPIcs.STACS.2022.48 000118000 0248_ $$2sideral$$a129136 000118000 037__ $$aART-2022-129136 000118000 041__ $$aeng 000118000 100__ $$aLutz, Jack H 000118000 245__ $$aExtending the Reach of the Point-To-Set Principle 000118000 260__ $$c2022 000118000 5060_ $$aAccess copy available to the general public$$fUnrestricted 000118000 5203_ $$aThe 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. 000118000 536__ $$9info:eu-repo/grantAgreement/ES/DGA/T64-20R$$9info:eu-repo/grantAgreement/ES/MICINN AEI/PID2019-104358RBI00$$9info:eu-repo/grantAgreement/ES/MICINN AEI/TIN2016-80347-R 000118000 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/ 000118000 592__ $$a0.832$$b2022 000118000 593__ $$aSoftware$$c2022 000118000 593__ $$aLogic$$c2022 000118000 655_4 $$ainfo:eu-repo/semantics/conferenceObject$$vinfo:eu-repo/semantics/publishedVersion 000118000 700__ $$aLutz, Neil 000118000 700__ $$0(orcid)0000-0002-9109-5337$$aMayordomo Cámara, Elvira$$uUniversidad de Zaragoza 000118000 7102_ $$15007$$2570$$aUniversidad de Zaragoza$$bDpto. Informát.Ingenie.Sistms.$$cÁrea Lenguajes y Sistemas Inf. 000118000 773__ $$g219 (2022), 48 [14 pp.]$$pLeibniz int. proc. inform.$$tLeibniz international proceedings in informatics$$x1868-8969 000118000 8564_ $$s735339$$uhttps://zaguan.unizar.es/record/118000/files/texto_completo.pdf$$yVersión publicada 000118000 8564_ $$s1994241$$uhttps://zaguan.unizar.es/record/118000/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000118000 909CO $$ooai:zaguan.unizar.es:118000$$particulos$$pdriver 000118000 951__ $$a2023-07-06-12:25:04 000118000 980__ $$aARTICLE