The class of $H^{\left(N\right)}$-sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{\left(N\right)}$-sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{\left(N\right)}$-sets.

We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology...