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

Abstract

We 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.

Regular variation without limits

“…A specific example may be constructed by appeal to Gödel's Axiom V = L and taking for the summands co-analytic Hamel bases; see [34, p. 256]. For further details of the vector sum see [46].…”

Dirichlet sets, Erdős-Kunen-Mauldin theorem, and analytic subgroups of the reals

“…The notion of perfectly meager sets has some natural modifications; see [10], [14]. Definition 4.4 (Zakrzewski).…”

The combinatorics of the Baer-Specker group

“…Proof. In [27] it is proved that if a set of reals has the Hurewicz property and has strong measure zero, then it has the Rothberger property (see [38] for a simple proof of that assertion). Use Theorem 3.5.…”