Cardinal invariants related to density

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak{i}<{\mathfrak{s}}_{1\mathrm{/}2}$), Con(${\mathfrak{r}}_{1\mathrm{/}2}<\mathfrak{b}$) and Con(${\mathfrak{i}}_{\ast }<2^{{\mathrm{\aleph }}_0}$). This answers two questions raised in arXiv:1808.02442v3. Besides, we prove the consistency of ${\mathfrak{s}}_{1\mathrm{/}2}^{\mathrm{\infty }}<$ non$(\mathcal{E})$ and cov$(\mathcal{E})<{\mathfrak{r}}_{1\mathrm{/}2}^{\mathrm{\infty }}$, where $\mathcal{E}$ is the $\sigma$-ideal generated by closed sets of measure zero.