The algebraic sum of sets of real numbers with strong measure zero sets

Abstract

AbstractWe 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 inis 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.