On a new class of trigonometric thin sets extending Arbault sets

Abstract

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.