Abstract 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.

Abstract We prove the following theorems: 1. IfX⊆ 2ωis aγ-set andY⊆2ωis a strongly meager set, thenX+Yis Ramsey null. 2. IfX⊆2ωis aγ-set andYbelongs to the class of sets, then the algebraic sumX+Yis an set as well. 3. Under CH there exists a setX∈MGR* which is not Ramsey null.

Abstract We prove the following theorems: (1) IfXhas strong measure zero and ifYhas strong first category, then their algebraic sum has property S0. (2) IfXhas Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. (3) IfXhas strong measure zero and Hurewicz's covering property then its algebraic sum with any set in is a set in . ( is included in the class of sets always of first category, and includes the class of strong first category sets.) These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the γ-property and of a first category set is a first category set, and Bartoszyfński and Judah's characterization of -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts.