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On the descriptive complexity of Salem sets

Volume 257 / 2022

Alberto Marcone, Manlio Valenti Fundamenta Mathematicae 257 (2022), 69-93 MSC: Primary 03E15; Secondary 28A75, 28A78, 03D32. DOI: 10.4064/fm997-7-2021 Published online: 15 November 2021

Abstract

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 requirement of 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 also holds for the Vietoris topology.

Authors

  • Alberto MarconeDepartment of Mathematics, Computer Science and Physics
    University of Udine
    33100 Udine, Italy
    e-mail
  • Manlio ValentiDepartment of Mathematics, Computer Science and Physics
    University of Udine
    33100 Udine, Italy
    e-mail

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