Topological Erdős similarity conjecture and strong measure zero sets
Yeonwook Jung, Chun-Kit Lai
TL;DR
The paper resolves the topological Erdős similarity variant by proving that a set is topologically universal on $\mathbb{R}$ if and only if it has strong measure zero; this result extends to locally compact Polish groups. It then leverages measure-category duality to formulate full measure universal sets and strongly meager sets, proposing dual conjectures and highlighting their independence from ZFC via the Borel conjecture. The authors show that the existence of uncountable topologically universal sets is independent of ZFC and provide a framework connecting affine transformations, Rothberger boundedness, and sumset properties through the Galvin-Mycielski-Solovay theorem. They also discuss open questions about perfect sets, the full-measure analogue, and potential generalizations, outlining a rich interaction between topology, measure, and group structure.
Abstract
We resolve the topological version of the Erdős 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 Erdős Similarity Conjecture with strongly meager sets via the duality of measure and category.