On the descriptive complexity of Salem sets

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 $oldsymbol{Pi}^0_3$-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 $oldsymbol{Pi}^0_3$-complete when we endow $mathbf{F}(mathbb{R}^d)$ with the Fell topology. A similar result holds also for the Vietoris topology. We apply our results to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions.