On the descriptive complexity of Salem sets
| Autor: | Alberto Marcone, Manlio Valenti |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Algebra and Number Theory
Degree (graph theory) Dimension (graph theory) Hausdorff space Dynamical Systems (math.DS) Mathematics - Logic Descriptive complexity theory Ambient space Combinatorics Compact space FOS: Mathematics 03E15 28A75 28A78 03D32 Family of sets Mathematics - Dynamical Systems Logic (math.LO) Mathematics Descriptive set theory |
| Zdroj: | Fundamenta Mathematicae. 257:69-93 |
| ISSN: | 1730-6329 0016-2736 |
| Popis: | 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. Comment: Extended Lemma 3.1, fixed Lemma 5.3 and improved the presentation of the results. To appear in Fundamenta Mathematicae |
| Databáze: | OpenAIRE |
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