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

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$(\mathcal{N})$).