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.
