# Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

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 $${\frak d}^0={\frak d}$$?

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?