Separating covering numbers and separating uniformity numbers of Hausdorff measures need large continuum

TL;DR

This paper demonstrates that under the assumption of the continuum being leph_2, all covering and uniformity numbers of Hausdorff measures in Euclidean spaces are equal, addressing specific open problems in measure theory.

Contribution

It shows that the covering and uniformity numbers of Hausdorff measures are equal under certain set-theoretic assumptions, providing partial solutions to open problems.

Findings

All covering numbers of Hausdorff measures are equal when leph_2 is the continuum.

All uniformity numbers of Hausdorff measures are equal under the same assumption.

The results depend on the set-theoretic assumption leph_2 = continuum.

Abstract

We show that if $\mathfrak{c} = \aleph_2$ then all covering numbers of Hausdorff measures $\operatorname{cov}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal and all uniformity numbers $\operatorname{non}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal. This is a partial answer to Problem 5.3 and 5.4 of [4].