论文标题
关于拓扑的相似性问题
On a topological Erdős similarity problem
论文作者
论文摘要
当集合中的每个集合都包含原始图案的一些线性和翻译副本时，在另一组集合中称为通用图案。保罗·埃尔德（Paul Erd）提出了一个猜想，即没有无限的集合在以积极度量的集合集合中是普遍的。本文探讨了拓扑设置中的一个类似问题。我们没有以积极的措施进行集合，而是研究了密集的$g_δ$集合的收集，并在通用集的集合中（密集的$g_δ$和补充的集合都具有lebesgue测量零）。我们将这种模式分别为普遍和普遍存在的模式。很容易证明任何可计数的集合在拓扑上都是通用的，而任何包含内部装饰的集合在拓扑上都不能是通用的。在本文中，我们将证明$ {\ mathbb r}^d $上的cantor set在拓扑上不是通用的，并且cantor套件在$ {\ mathbb r}^1 $上具有正纽舍厚度，并非通用。这给出了一个关于问题的积极部分答案，该问题涉及Cantor集中的Erdő相似性问题。此外，我们还获得了通用普遍性问题的更高维度的概括。
A pattern is called universal in another collection of sets, when every set in the collection contains some linear and translated copy of the original pattern. Paul Erdős proposed a conjecture that no infinite set is universal in the collection of sets with positive measure. This paper explores an analogous problem in the topological setting. Instead of sets with positive measure, we investigate the collection of dense $G_δ$ sets and in the collection of generic sets (dense $G_δ$ and complement has Lebesgue measure zero). We refer to such pattern as topologically universal and generically universal respectively. It is easy to show that any countable set is topologically universal, while any set containing an interior cannot be topologically universal. In this paper, we will show that Cantor sets on ${\mathbb R}^d$ are not topologically universal and Cantor sets with positive Newhouse thickness on ${\mathbb R}^1$ are not generically universal. This gives a positive partial answer to a question by Svetic concerning the Erdős similarity problem on Cantor sets. Moreover, we also obtain a higher dimensional generalization of the generic universality problem.
