Executive Summary : | The notion of characterized subgroups has evolved over the years as a generalization of the notion of {torsion subgroup} (recall that an element $x$ of an abelian group is torsion if there exists $k \in \mathbb{N}$ such that $kx = 0$). An element $x$ of an abelian topological group $G$ is called (i) { topologically torsion} if $n!x \rightarrow 0;$ (ii) {topologically $p$-torsion}, for a prime $p$, if $p^nx \rightarrow 0.$ In recent investigations it has been seen that one can generate versions of characterized subgroups which are in general larger in size than their classical counterparts by using more general notions of convergence which arise naturally for density functions or more generally for ideals. It has already been established that these newly formed subgroups are essentially different and strictly larger in size than the much investigated class of characterized subgroups, having cardinality c but remaining nontrivial (i.e. different from T) mostly remaining Fσδ. There are several open problems and lines of investigations in this direction of which we intend to explore in this project. The characterized subgroups of the circle group on the other hand forms a basis of Arbault sets, an important class of thin sets occurring in harmonic analysis. It has been observed that there are statistically characterized subgroups which can't be characterized by any sequence of integers establishing the ``newness" of the notion. 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. Similar classes of thin sets come naturally for generalized classes of characterized subgroups which we further intend to study along with their possible roles in harmonic analysis. |