PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 137–144 S 0002-9939(97)03516-8 A FAMILY OF PERMITTED TRIGONOMETRIC THIN SETS MIROSLAV REPICK ´ Y (Communicated by Andreas R. Blass) Abstract. We introduce the notion of perfectly measure zero sets and prove that every perfectly measure zero set is permitted for the families of all pseudo- Dirichlet sets, N 0 -sets, A-sets and N-sets. In particular this means that these families of trigonometric thin sets are closed under adding sets of cardinality less than the additivity of Lebesgue measure. §0. Introduction Let F be a family of sets of reals. Let A, B ∈F . We say that a set A is F - permitted for B if A ∪ B ∈F . A set A is F -permitted if A is F -permitted for every B ∈F (see [1]). Let A be set of reals. Let us recall the following notions. (1) A is a pD-set (pseudo-Dirichlet set) if there is an increasing sequence of inte- gers {n k } ∞ k=0 such that the sequence {sin n k πx} ∞ k=0 converges quasinormally on A, i.e. there is a sequence of positive reals {ε k } ∞ k=0 converging to 0 such that (∀x ∈ A)(∀ ∞ k) | sin n k πx| <ε k . (2) A is an N 0 -set if there is an increasing sequence of integers {n k } ∞ k=0 such that ∑ ∞ k=0 | sin n k πx| < ∞ for x ∈ A. (3) A is an A-set if there is an increasing sequence of integers {n k } ∞ k=0 such that {sin n k πx} ∞ k=0 converges to 0 for x ∈ A. (4) A is an N-set if there is a sequence of non-negative reals {ρ n } ∞ n=0 such that ∑ ∞ n=0 ρ n = ∞ and the series ∑ ∞ n=0 |ρ n sin nπx| converges for x ∈ A. The families of all pD-sets, N 0 -sets, A-sets, and N-sets are denoted by pD, N 0 , A, and N , respectively. An Arbault-Erd¨ os theorem [1] says that every countable set is N -permitted. N. N. Kholshchevnikova [11] proved that under Martin’s Axiom every set of car- dinality less than c is N -permitted. L. Bukovsk´ y and Z. Bukovsk´ a [6] proved that every set of cardinality less than p is N -permitted. These results have been improved [7] by showing that every γ -set is permitted for all families of thin sets de- fined above. In the present paper we prove that every set having perfectly measure zero (Definition 1.1) is permitted for these families. This improves the previously mentioned results since every γ -set has perfectly measure zero. Since every set Received by the editors February 10, 1995 and, in revised form, June 6, 1995. 1991 Mathematics Subject Classification. Primary 42A20; Secondary 03E05, 03E20. Key words and phrases. Trigonometric thin sets, permitted sets, perfectly measure zero sets, cardinal invariants. Thework has been supported by grant2/1224/94 of Slovensk´a grantov´a agent´ ura. c 1997 American Mathematical Society 137 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use