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Published online by Cambridge University Press: 02 September 2025
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(
$\mathfrak {i}<\mathfrak {s}_{1/2}$), Con( $\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con(
$\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con( $\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of
$\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of  $\mathfrak {s}_{1/2}^{\infty } < $ non
$\mathfrak {s}_{1/2}^{\infty } < $ non $(\mathcal {E})$ and cov
$(\mathcal {E})$ and cov $(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where
$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where  $\mathcal {E}$ is the
$\mathcal {E}$ is the  $\sigma $-ideal generated by closed sets of measure zero.
$\sigma $-ideal generated by closed sets of measure zero.