Combinatorics of translations of meager and closed measure zero sets

We study combinatorial properties of translations of meager and closed sets of measure zero. We discuss constellations of Borel conjectures for related classes of small sets and show that there are no uncountable null-additive sets in the Miller model. We also study cardinal invariants of those classes. In particular, we investigate the cardinal invariants of $\sigma$-ideals ${\mathcal{H}}_F$ related to meager-additive sets. We obtain a new characterization of additivity of meager ideal and answer some questions of Cardona. We also show that $\mathrm{n}\mathrm{o}\mathrm{n}({\mathcal{E}}^{\ast })$, that is equal to the translation version of the covering number of the ideal $\mathcal{E}$, is close to the cardinal invariant related to the Laver property. This strengthens a result of Bartoszyński and Judah and a result of Elekes and Steprāns. Finally, we show that every $\mathcal{E}$-Luzin set is in ${\mathcal{E}}^{\ast }$.