Effective aspects of Hausdorff and Fourier dimension
Abstract
In this paper, we study Hausdorff and Fourier dimension from the point of view of effective descriptive set theory and Type2 Theory of Effectivity. Working in the hyperspace $\mathbf{K}(X)$ of compact subsets of $X$, with $X=[0,1]^d$ or $X=\mathbb{R}^d$, 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 $\Pi^0_3$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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.06941
 Bibcode:
 2021arXiv210806941M
 Keywords:

 Mathematics  Logic;
 Mathematics  Functional Analysis;
 Primary: 03D78;
 Secondary: 03D55;
 28A75;
 28A78
 EPrint:
 36 pages