Antistable classes of thin sets in harmonic analysis (Q1320922)

In this paper we study several classes of thin sets in Harmonic Analysis, like sets of absolute convergence, compact Dirichlet sets, Arbault sets or \(H\)-sets. In the first part we make precise the relationships between these classes. In particular, we prove the existence of a set of resolution which is not a set of absolute convergence, solving a problem of N. Bary. In the second part we study the stability of these classes under various set-theoretic operations (like finite union or increasing countable union), and prove that they are as far from being stable as possible. This in particular solves a question of J. Arbault about sets of absolute convergence, but also provides a general tool for proving such instability results.