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

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 sX, rX and ddX,Y introduced in arXiv:2304.09698 and arXiv:2410.21102, answering some questions raised in these papers. In particular, we prove that s₀= cov(M) and r₀= non(M). We also show that dd_\, all=dd_\1/2\, all for all r∈ (0,1), and we provide a proof of Con(dd_(0,1),\0,1\ʳᵉˡ< non(N)) and Con(ddall,allʳᵉˡ< non(N)).