Splitting Families, Reaping Families, and Families of Permutations Associated with Asymptotic Density

TL;DR

This paper explores relationships between various cardinal characteristics of the continuum related to asymptotic density, establishing equalities and inequalities among them and connecting them to known invariants like cov$( ext{meager})$ and non$( ext{meager})$.

Contribution

It identifies new equalities and inequalities among cardinal invariants associated with asymptotic density, answering open questions from prior research.

Findings

Proves $rak{s}_0=$ cov$( ext{meager})$ and $rak{r}_0=$ non$( ext{meager})$.

Shows $rak{dd}_{ , ext{all}}=rak{dd}_{rac{1}{2}, ext{all}}$ for all $r eq 0$.

Establishes consistency results involving $rak{dd}$ and non$( ext{null})$.

Abstract

We investigate several relations between cardinal characteristics of the continuum related with the asymptotic density of the natural numbers and some known cardinal invariants. Specifically, we study the cardinals of the form $\mathfrak{s}_X$, $\mathfrak{r}_X$ and $\mathfrak{dd}_{X,Y}$ introduced in arXiv:2304.09698 and arXiv:2410.21102, answering some questions raised in these papers. In particular, we prove that $\mathfrak{s}_0=$ cov$(\mathcal{M})$ and $\mathfrak{r}_0=$ non$(\mathcal{M})$. We also show that $\mathfrak{dd}_{\{r\}, \textsf{all}}=\mathfrak{dd}_{\{1/2\}, \textsf{all}}$ for all $r\in (0,1)$, and we provide a proof of Con($\mathfrak{dd}_{(0,1),\{0,1\}}^{\textsf{rel}}<$ non$(\mathcal{N})$) and Con($\mathfrak{dd}_{\textsf{all},\textsf{all}}^{\textsf{rel}}<$ non$(\mathcal{N})$).