A hierarchy of thin sets related to the boundedness of trigonometric series

Abstract:

We study the family $\mathcal {B}_0$ of the sets on which some series of the form $\sum _{k\in \mathbb {N}}\left |\sin \pi n_kx\right |$ is uniformly bounded. We show that the families $\mathcal {B}_0^c$ of all sets admitting the boundary $c$ form a hierarchy which is incontinuous with respect to the operations of intersection and union.