Cardinal invariants related to density
David Valderrama
TL;DR
The paper investigates density-based variants of splitting and reaping defined via asymptotic density, focusing on invariants $\mathfrak{s}_{1/2}$, $\mathfrak{r}_{1/2}$, the independence number $\mathfrak{i}$, and the $*$-independence $\mathfrak{i}_{*}$, and their relation to the $\sigma$-ideal $\mathcal{E}$. It develops $\sigma$-centered forcing methods to preserve density-based cardinals and employs Hechler forcing, Cohen forcing, and Mathias forcing to separate these invariants, establishing Con$(\mathfrak{d}<\mathfrak{s}_{1/2})$, Con$(\mathfrak{r}_{1/2}<\mathfrak{b})$, and Con$(\mathfrak{i}_{*}<2^{\aleph_0})$, along with $\mathfrak{s}_{1/2}^{\infty} \leq \text{non}(\mathcal{E})$ and $\text{cov}(\mathcal{E}) \leq \mathfrak{r}_{1/2}^{\infty}$. It also demonstrates the consistency of strict inequalities involving $\mathcal{E}$ via Hechler and Dual Hechler models, and proves Con$(\mathfrak{i}_{*}<2^{\aleph_0})$ in a Cohen-model under CH. Together, these results extend density-based invariants beyond classical splitting/reaping and clarify their interplay with the ideal $\mathcal{E}$, while showcasing robust forcing techniques to separate these cardinals.
Abstract
We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak{i}<\mathfrak{s}_{1/2}$), Con($\mathfrak{r}_{1/2}<\mathfrak{b}$) and Con($\mathfrak{i}_*<2^{\aleph_0}$). This answers two questions raised in arXiv:1808.02442v3. Besides, we prove the consistency of $\mathfrak{s}_{1/2}^{\infty} < $ non$(\mathcal{E})$ and cov$(\mathcal{E}) < \mathfrak{r}_{1/2}^{\infty}$, where $\mathcal{E}$ is the $σ$-ideal generated by closed sets of measure zero.