论文标题
关于塞勒姆集的描述性复杂性
On the descriptive complexity of Salem sets
论文作者
论文摘要
在本文中，我们从描述集理论的角度研究了塞勒姆集的概念。我们首先在$ [0,1] $的紧凑子集的Hyperspace $ \ Mathbf {K}（[0,1]）$中工作，并表明已封闭的塞勒姆集合形成了$ \boldsymbolπ^0_3 $ -complete family。这是通过表征具有足够大的Hausdorff或傅立叶维度的集合家族的复杂性来完成的。我们还表明，如果我们增加了环境空间的维度并在$ \ mathbf {k}（[0,1]^d）$中工作，则复杂性不会改变。然后，我们通过放松环境空间的紧凑性来概括结果，并表明封闭的塞勒姆集仍然是$ \boldsymbolπ^0_3 $ -complete，当我们赋予所有$ \ m athbb {r}^d $的超级空间。越野拓扑也有类似的结果。
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 $\boldsymbolΠ^0_3$-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 $\boldsymbolΠ^0_3$-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.
