Topological Erd\H{o}s similarity conjecture and strong measure zero sets

TL;DR

This paper proves that a set in the real line is topologically universal if and only if it has strong measure zero, linking topological universality to measure-theoretic properties and exploring independence results in set theory.

Contribution

It establishes the equivalence between topological universality and strong measure zero sets, and discusses independence of uncountable universal sets from ZFC, extending results to Polish groups.

Findings

Topologically universal sets are exactly the strong measure zero sets.

Existence of uncountable topologically universal sets is independent of ZFC.

Results extend to locally compact Polish groups.

Abstract

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a result of the fact that the Borel conjecture is independent of the \textsf{ZFC} axiomatic set theory, the existence of an uncountable topologically universal set is independent of the \textsf{ZFC}. Moreover, our results can also be generalized to locally compact Polish groups ${\mathbb G}$. Returning to the measure side, we pose Full-Measure universal Erd\H{o}s Similarity Conjecture with strongly meager sets via the duality of measure and category.