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 Type-2 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 ${\Pi }_3^0$-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.

36 pages

References in corpus (4)

• A comparison of concepts from computable analysis and effective descriptive set theory
• The descriptive theory of represented spaces
• Explicit Salem sets in ${\mathbb{R}}^n$
• On the descriptive complexity of Salem sets

Cited by in corpus (1)

• A jump operator on the Weihrauch degrees