Let
be a countable
multiset of primes and let
.
We study the universal characteristic factors associated with the Gowers–Host–Kra seminorms
for the group
.
We show that the universal characteristic factor of order
is a factor of an inverse limit of
finite-dimensional-stepnilpotent homogeneous spaces. The latter is a counterpart of a
-step
nilsystem where the homogeneous group is not necessarily a Lie group. As an
application of our structure theorem we derive an alternative proof for the
-convergence
of multiple ergodic averages associated with
-term arithmetic
progressions in
and derive a formula for the limit in the special case where the underlying space is a
nilpotent homogeneous system. Our results provide a counterpart of the
structure theorem of Host and Kra (2005) and Ziegler (2007) concerning
-actions
and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning
-actions.
This is also the first instance of studying the Host–Kra factors of nonfinitely
generated groups of unbounded torsion.
