Effective aspects of Hausdorff and Fourier dimension

Issue title: Oberwolfach Workshop on Computability Theory 2021

Guest editors: Vasco Brattka, Noam Greenberg, Iskander Kalimullin and Mariya Soskova

Article type: Research Article

Authors: Marcone, Alberto^{a; **} | Valenti, Manlio^{b; *; **}

Affiliations: [a] Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy | [b] Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Correspondence: [**] Corresponding authors. alberto.marcone@uniud.it; manliovalenti@gmail.com.

Note: [*] Current address: Department of Mathematics, Univerisity of Wisconsin – Madison, Madison, WI, USA

Abstract: In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity. Working in the hyperspace K(X) of compact subsets of X, with X=[0,1]d or X=Rd, we characterize the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. This, in turn, allows us to show that family of all the closed Salem sets is Π30-complete. One of our main tools is a careful analysis of the effectiveness of a classical theorem of Kaufman. We furthermore compute the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.

Keywords: Computable analysis, Hausdorff dimension, Fourier dimension, Salem sets

DOI: 10.3233/COM-210372

Journal: Computability, vol. 11, no. 3-4, pp. 299-333, 2022