On a topological Erd\H{o}s similarity problem

TL;DR

This paper investigates topological analogs of Erdős similarity problems, showing that certain Cantor sets are not universal in dense G_delta or generic sets, thus extending the classical measure-based conjectures to a topological context.

Contribution

It introduces the concepts of topologically and generically universal sets, and proves that specific Cantor sets are not universal in these topological categories, partially answering Erdős similarity conjecture.

Findings

Cantor sets on R^d are not topologically universal.

Cantor sets with positive Newhouse thickness are not generically universal.

Provides a higher-dimensional generalization of the generic universality problem.

Abstract

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 Erd\H{o}s 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 $G_{\delta}$ sets and in the collection of generic sets (dense $G_{\delta}$ 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 Erd\H{o}s similarity problem on Cantor sets. Moreover, we also obtain a higher dimensional generalization of the generic universality problem.