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