Cardinal invariants associated with Hausdorff measures

TL;DR

This paper investigates cardinal invariants related to Hausdorff measures, demonstrating their separation in various models and establishing results about uniformity numbers for specific measure ideals.

Contribution

It introduces new separations of cardinal invariants associated with Hausdorff measure zero ideals using forcing models and analyzes their uniformity numbers.

Findings

Separation of many cardinal invariants of Hausdorff measure zero ideals.

Uniformity numbers of s-dimensional Hausdorff measure zero ideals for 0<s<1.

Separation of Lebesgue null ideal's uniformity number using Mathias forcing.

Abstract

We consider cardinal invariants determined from Hausdorff measures. We separate many cardinal invariants of Hausdorff measure $0$ ideals using two models that separate many cardinal invariants of Yorioka ideals at once from earlier work. Also we show the uniformity numbers of $s$-dimensional Hausdorff measure $0$ ideals for $0 < s < 1$ and that of Lebesgue null ideal can be separated using the Mathias forcing.