When ideals properly extend the class of Arbault sets

Abstract

In this article we continue the investigation of generalized version of Arbault sets, that was initiated in [Das et al., Bul. Sci. Math. 179 (2022), 103157] but look at the picture from the most general point of view where ideals come into play. While Arbault sets can be naturally associated with the Frechet ideal $Fin$, in [Das et al., Bul. Sci. Math. 179 (2022), 103157] it was observed that when $Fin$ is replaced by the natural density ideal $\iI_d$ one can obtain a strictly larger class of trigonometric thin sets containing Arbault sets. From the set theoretic point of view a natural question arises as to whether one can broaden the picture and specify a class of ideals (instead of a single ideal) each of which would have the similar effect on the classical notion. As a natural candidate, we focus on a special class of ideals, namely, non-$snt$ ideals with a specific property ($snt$ stands for ``strongly non translation invariant"). This class happens to be quite large and rich as it properly contains the class of all dense translation invariant ideals ($\varsupsetneq Fin$), ideals generated by simple density functions as also certain non-negative regular summability matrices (but not all) which can be seen from [Das et al., Annals of Pure and Applied Logic 174 (2023), 103289]. We consider the resulting class of $\iI$-Arbault sets and it is observed that for each such ideal, the class of $\iI$-Arbault sets not only properly contains the class of classical Arbault sets but also a large subfamily of $\NN$-sets (also called ``sets of absolute convergence") while being contained in the class of weak Dirichlet sets.