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

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $\unicode{x3b3} $, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $\unicode{x3b3} $ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin–Miller and Bartoszyński–Recław, to obtain sets with analogous properties. We also consider products of Sierpiński sets in the context of combinatorial covering properties.