### The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets

Andrej Nowik, Marion Scheepers, and Tomasz Weiss
Source: J. Symbolic Logic Volume 63, Issue 1 (1998), 301-324.

#### Abstract

We prove the following theorems: (1) If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s$_0$. (2) If X has 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) If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in $\mathcal{APC}$ ' is a set in $\mathcal{APC}$ '. ($\mathcal{APC}$ ' 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 $\gamma$-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR$^\mathcal{M}$-sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts.

Primary Subjects: 03E20
Secondary Subjects: 28E15, 54F65, 54G99
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