NULL SETS AND COMBINATORIAL COVERING PROPERTIES

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