We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$

Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering relation $\leq^0$ on $\omega^\omega$ by saying that $f\leq^0 g$ if $f(x) \leq g(x)$ for all $x\in\omega\setminus N$ where $N\subseteq \omega$ is a nullset.

Similarly to the bounding number and the dominating number respectively, we define

${\frak b}^0 = \min\{|B|: B\subseteq \omega^\omega \land \forall f\in \omega^\omega\; \exists b\in B(b\not\leq^0 f)\}$, and

${\frak d}^0 = \min\{|D|: D\subseteq \omega^\omega \land \forall f\in \omega^\omega\; \exists d\in D(f\leq^0 d)\}$.

Do we have ${\frak b}^0={\frak b}$? And what about ${\frak d}^0={\frak d}$?


The answer seems to be positive according to this paper: Barnabás Farkas, Lajos Soukup: The zero density ideal, cardinal invariants and related forcing problems.1

Theorem 2.3. If $\mathcal I$ is a rare ideal on $\mathbb N$ then $\mathfrak b = \mathfrak b_{\mathcal I}$ and $\mathfrak d = \mathfrak d_{\mathcal I}$.

Just before this theorem the authors mention that the ideal $\mathcal Z_0$ of the sets with zero density is a rare ideal.

A similar result is shown for analytic P-ideals in Corollary 5.5 of More on cardinal invariants of analytic P-ideals by the same two authors (arXiv, eudml). Again, this class of ideals includes $\mathcal Z_0$.

1I wasn't able to find whether the paper was published somewhere, but a preprint is available here (Wayback Machine). The same paper was also mentioned in this answer: Are these two quotients of $\omega^\omega$ isomorphic?


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