Towers and permitted trigonometric thin sets

Abstract

In \cite{R} we introduced the notion of perfect measure zero sets and proved that every perfect measure zero set is permitted for any of the four families of trigonometric thin sets \(\\mathcal{N}\), \(\mathcal{A}\), \(\mathcal{N}_0\), and \(p\mathcal{D}\). Now we prove that the unions of less than \(\mathcal{t}\) perfect measure zero sets are permitted for the mentioned families. This strengthens a result of T. Bartoszyński and M. Scheepers \cite{BSch} saying that every set of cardinality less than \(\mathcal{t}\) is \(\mathcal{N}\)-permitted.