THE ALGEBRAIC SUM OF SETS OF REAL-NUMBERS WITH STRONG MEASURE ZERO-SETS

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 me asure 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 Hure wicz's covering property then its algebraic sum with any set in AFC' i s a set in AFC'. (AFC' is included in the class of sets always of firs t category, and includes the class of strong first category sets.) The se results extend: Fremlin and Miller's theorem that strong measure ze ro sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the gamma-pr operty and of a first category set is a first category set, and Bartos zynski and Judah's characterization of SRM-sets. They also characteriz e the property () introduced by Gerlits and Nagy in terms of older co ncepts.