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](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum#Bounding_number_'"<code>UNIQ--postMath-00000015-QINU</code>"'_and_dominating_number_'"<code>UNIQ--postMath-00000016-QINU</code>"') and the [dominating number](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum#Bounding_number_'"<code>UNIQ--postMath-00000015-QINU</code>"'_and_dominating_number_'"<code>UNIQ--postMath-00000016-QINU</code>"') 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 ${\frak d}^0={\frak d}$?



Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets
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](https://arxiv.org/abs/1002.2192), [eudml](https://eudml.org/doc/32499)). 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](https://www.renyi.hu/~descript/papers/Fa_So_zero.pdf) ([Wayback Machine](http://web.archive.org/web/20120119124441/https://www.renyi.hu/~descript/papers/Fa_So_zero.pdf)). The same paper was also mentioned in this answer: [Are these two quotients of $\omega^\omega$ isomorphic?](https://mathoverflow.net/q/231077#231088)


