Effective aspects of Hausdorff and Fourier dimension

TL;DR

This paper explores the complexity and computability of Hausdorff and Fourier dimensions within effective descriptive set theory, revealing the classification of Salem sets and analyzing the effectiveness of classical theorems.

Contribution

It characterizes the complexity of sets with large Hausdorff or Fourier dimension and computes the Weihrauch degrees of dimension functions, advancing the understanding of their effective properties.

Findings

The family of all closed Salem sets is $oldsymbol{ ext{Pi}}^0_3$-complete.

Provides a detailed analysis of the effectiveness of Kaufman's theorem.

Computes the Weihrauch degree of functions for Hausdorff and Fourier dimensions.

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 $\Pi^0_3$-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.