On the descriptive complexity of Salem sets

TL;DR

This paper investigates the descriptive set-theoretic complexity of Salem sets, showing they form a highly complex class within various hyperspaces, regardless of ambient space dimension or topology.

Contribution

It characterizes the complexity of Salem sets as a $oldsymbol{oldsymbol{ ext{Pi}}}^0_3$-complete family in multiple hyperspace topologies, extending previous understanding.

Findings

Salem sets form a $oldsymbol{ ext{Pi}}^0_3$-complete family in hyperspaces.

Complexity remains the same across different ambient space dimensions.

Results hold for various topologies including Fell and Vietoris.

Abstract

In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol{\Pi}^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $\boldsymbol{\Pi}^0_3$-complete when we endow the hyperspace of all closed subsets of $\mathbb{R}^d$ with the Fell topology. A similar result holds also for the Vietoris topology.