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

We study the family  $\mathcal{B}_0$ of the sets on which some series of the form $\sum_{k\in{\Bbb N}}\left\vert\sin\pi n_kx\right\vert$ 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.