When ideals properly extend the class of Arbault sets
arXiv:2401.02103
The paper studies how replacing the Fréchet ideal Fin with other ideals, especially non‑snt ideals, expands the class of Arbault (trigonometric thin) sets, showing these new I‑Arbault sets strictly contain classical Arbault sets and many absolute convergence sets while staying within weak Dirichlet 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 , in [Das et al., Bul. Sci. Math. 179 (2022), 103157] it was observed that when 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- ideals with a specific property ( 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 (), 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.