An introduction to I-characterized subgroups of the circle

A subgroup H of the circle group T is said to be characterized by a sequence u (u_n)_{n in N} of integers if H={x in T: u_n x converges to 0}. The introduction of these subgroups was motivated by problems arising from various areas of Mathematics, so they were thoroughly investigated. Recently generalizations of this notion were introduced based on weaker notions of convergence, starting from statistical convergence and ending with I-convergence for an ideal I of N. This survey paper is dedicated to collect the wealth of results and open problems obtained on these new kind of characterized subgroups of T.