Perhaps someone count direct me to some experts and/or references?

Let *Z be an ultrapower of the integers Z, with respect to a
nonprincipal ultrafilter on a countably infinite index set.

For any real number r, one can associate a subgroup of *Z:

S_r = { n in *Z:    nr \in *Z + I }

where   I \subset *R  consists of  the infinitesimals in the
corresponding ultrapower *R of R.

Question:

To what extent does the structure of S_r depend on r?

For example, are any two such subgroups (for irrational values of r)
always isomorphic, or not? Is this easy to see from structure theory
of infinite abelian groups?