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 s0. 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 AFC ′ is a set in AFC . (AFC ′ 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 Bartoszyński and Judah’s characterization of SR–sets. They also characterize the property (∗) introduced by Gerlits and Nagy in terms of older concepts. 3 4 According to Borel a set of real numbers has property C if there is for every sequence (ǫn : n = 1, 2, 3, . . .) of positive real numbers a partition of the set into countably many pieces in such a way that for each n the n–th piece has diameter at most ǫn – [3]. Most authors nowadays call property C strong measure zero. Galvin, Mycielski and Solovay have shown that a set X of real numbers has strong measure zero if, and only if, for every first category set M there is a real number not belonging to the set X +M (= {x+m : x ∈ X and m ∈ M}) – [8]. X is said to be meager–additive if for every first category set M the set X + M is a first category set. In light of the Galvin–Mycielski–Solovay theorem every meager–additive set is a strong measure zero set. Borel conjectured that only countable sets have strong measure zero. Since Laver showed that Borel’s conjecture is not disprovable (if classical mathematics contains no contradictions) – [11] – one cannot prove that the notions of meager additive and Supported by KBN grant 2 PO3A 047 09 Partially supported by NSF grant DMS 95-05375