Remarks on sets related to trigonometric series (Q1896626)

The paper is a continuation of the research started by J. Arbault in the 1950s and later using modern methods of set theory (Martin's Axiom and forcing) by N. N. Kholshchevnikova, L. Bukovský and Z. Bukovská. For a given family of thin sets of reals (which is not an ideal) it is a natural question what is the minimal size \(\kappa\) of a set of reals whose union with a member of the family does not belong to the family. The authors found new lower bounds of the cardinal \(\kappa\) for several families of sets related to the convergence of trigonometric series, namely, \(\kappa\geq {\mathfrak t}\) for \(N\)-sets, \(\kappa\geq {\mathfrak h}\) for \(N_0\)-sets and \(\sigma\)-Dirichlet sets, and \(\kappa\geq {\mathfrak s}\) for \(A\)-sets and \(\sigma\)-Hardy sets (the cardinals \(\mathfrak h\), \(\mathfrak t\), \(\mathfrak s\) are the well-known cardinal invariants of the algebra \({\mathcal P}(\omega)\)/finite). -- Let us remark a terminology correction: instead of ``\(\sigma\)-Dirichlet'' the authors say ``pseudo Dirichlet'' which in the literature marks a somewhat different notion of thin sets.