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.

We prove the following theorems:1. IfX⊆ 2ωis aγ-set andY⊆2ωis a strongly meager set, thenX+Yis Ramsey null.2. IfX⊆2ωis aγ-set andYbelongs to the class ofsets, then the algebraic sumX+Yis anset as well.3. Under CH there exists a setX∈MGR* which is not Ramsey null.

Abstract. We prove the following theorems:1. It is consistent with ZFC that there exists a Q -set which is not perfectly meager in the transitive sense. 2. Every set which is perfectly meager in the transitive sense has the AF C property. 3. The product of two sets perfectly meager in the transitive sense has also that property.In this part we prove that it is consistent with ZFC that there are uncountable Q -sets which are not perfectly meager in the transitive sense. Most of the lemmas needed to show this result are based on I. Rec law [R], and H. Judah, S. Shelah [JS]. Throughout the proof, terminology and notations from the two papers mentioned above are being used.We use the following definition (see [NSW]) of sets that are perfectly meager in the transitive sense. Definition 1. Let X be a subset of the real line (or 2 ω , respectively). We say that X is an AF C set (perfectly meager in the transitive sense) iff for every perfect set D ⊆ R (D ⊆ 2 ω , respectively), one can find F , an F σ set containing X, such that for every t ∈ R (2 ω ), (F + t) ∩ D is meager in the relative topology of D.In [NSW], it is shown that (assuming Martin's Axiom) the class AF C is strictly included in the class AF C of perfectly meager sets.Let us also recall that a set X ⊆ R (or 2 ω , respectively) is called a Q -set iff its every subset is an F σ set in the relative topology of X. It is well -known that every Q -set is perfectly meager (see for example [M]). Proof. See Lemma 1 in [R].

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