On a new class of trigonometric thin sets extending Arbault sets

In this article we introduce a new class of trigonometric thin sets, namely statistical Arbault sets properly containing the class of classical Arbault sets [1] as well as a large subfamily of N-sets (also called "sets of absolute convergence ") [17]. In particular this class happens to contain the types of N-sets (that includes again strictly the so called N0-sets) which have been extensively used in the literature (see [1,23,24]), but at the same time being distinct from the class of N-sets. We observe that this is a new class strictly lying between the class of Arbault sets and the class of Weak Dirichlet sets. The motivation for thinking of such a possible class comes from the well known observation that the characterized subgroups of the circle group forms a basis of Arbault sets and the recent introduction of the notion of statistically characterized subgroups [14] extending the notion of characterized subgroups. In Section 2 of the article it is shown that there are statistically characterized subgroups which can't be characterized by any sequence of integers establishing the "novelty " of the notion. Other significance of this result is that it presents an example of a polishable subgroup of the circle group that can not be characterized by a suitable sequence of integers unlike the countable case whose answer is positive [5]. This naturally paves the way for a new class of sets generated by the class of statistically characterized subgroups as basis which we name statistical Arbault sets. (C) 2022 Elsevier Masson SAS. All rights reserved.