Uniformity numbers of the null-additive and meager-additive ideals

Denote by $\mathcal{N}\mathcal{A}$ and $\mathcal{M}\mathcal{A}$ the ideals of null-additive and meager-additive subsets of~$2^{\omega }$, respectively. We prove in ZFC that $\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{N}\mathcal{A})=\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{N}\mathcal{A})$ and introduce a new (Polish) relational system to reformulate Bartoszyński’s and Judah’s characterization of the uniformity of $\mathcal{M}\mathcal{A}$, which is helpful to understand the combinatorics of $\mathcal{M}\mathcal{A}$ and to prove consistency results. As for the latter, we prove that $\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M}\mathcal{A})<\mathfrak{c}$ (even $\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M}\mathcal{A})<\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{N})$) is consistent with ZFC, as well as several constellations of Cichoń’s diagram with $\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{N}\mathcal{A})$, $\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{M}\mathcal{A})$ and $\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{S}\mathcal{N})$, which include $\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{N}\mathcal{A})<\mathfrak{b}<\mathrm{n}\mathrm{o}\mathrm{n}(\mathcal{M}\mathcal{A})$ and $\mathfrak{b}<\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{S}\mathcal{N})<\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$.