Let T = R / Z be the written additively circle group and u =
(un) be a sequence of integers. Many authors in various areas
of Mathematics gave their attention to the following subgroups
of T and their subsets
t u( T ) = { x ∈ T | unx → 0 } .
These subgroups are known with various names, here I refer
to these subgroups as topologically u-torsion subgroups, because
of their strong connection with torsion subgroups. Here, be-
sides these subgroups in the circle group, I consider their nat-
ural generalization for an arbitrary topological abelian group,
with particular attention to the compact case: for a topologi-
cal abelian group X and a sequence of characters v = (vn) the
following subgroup
s v(X) = { x ∈ X | vn(x) → 0 }
is called characterized subgroup.
Here I present some of my research results. In particular, I
give a complete description of the subgroups t u( T ) where u
is an arithmetic sequence, that is a strictly increasing sequence
where un | un+1 for every n ∈ N. I give also some new results
on the study of the Borel complexity of these subgroups, both
in the compact case and in the circle group. Moreover, I present
a structure theorem for the subgroups that admit a finer locally
compact Polish group topology. The latter is a sufficient condi-
tion for a subgroup to be characterized. Furthermore, I give a
complete description of closed characterized subgroups in arbi-
trary topological abelian groups and various useful reductions
to the metrizable case. Presenting these results, I take the op-
portunity to give an exhaustive description of the state of the
art in this topic and to show some applications to other areas of
Mathematics, with the aim of providing a useful handbook to
an expert audience and a starting point for potential research
purposes to non-expert users.
