论塞勒姆集合描述的复杂性

Pub Date : 2020-09-21 DOI:10.4064/fm997-7-2021
A. Marcone, Manlio Valenti
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引用次数: 1

摘要

本文从描述集合论的角度研究了Salem集合的概念。我们首先在$[0,1]$的紧子集$\mathbf{K}([0,1])$的超空间$\mathbf{K}([0,1])$上进行研究,证明了闭塞勒姆集形成$\boldsymbol{\Pi}^0_3$-完备族。这是通过描述具有足够大的豪斯多夫维数或傅里叶维数的集合族的复杂性来实现的。我们还表明,如果我们增加环境空间的维度并在$\mathbf{K}([0,1]^d)$中工作,复杂度不会改变。然后我们通过放松环境空间的紧性来推广结果,并证明当我们赋予$\mathbf{F}(\mathbb{R}^d)$具有Fell拓扑时,闭Salem集仍然是$\boldsymbol{\Pi}^0_3$-完备的。类似的结果也适用于Vietoris拓扑。我们应用我们的结果来表征计算豪斯多夫维数和傅立叶维数的函数的魏氏度。
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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 $\boldsymbol{\Pi}^0_3$-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 $\mathbf{F}(\mathbb{R}^d)$ 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.
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