Host-Kra factors for $\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$ actions and finite dimensional nilpotent systems

Let $\mathcal{P}$ be a countable multiset of primes and let $G=\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$. We study the universal characteristic factors associated with the Gowers-Host-Kra seminorms for the group $G$. We show that the universal characteristic factor of order $<k+1$ is a factor of an inverse limit of finite dimensional $k$-step nilpotent homogeneous spaces. The latter is a counterpart of a $k$-step nilsystem where the homogeneous group is not necessarily a Lie group. This result provides a counterpart of the structure theorem of Host-Kra and Ziegler concerning $\mathbb{Z}$-actions and generalizes the results of Bergelson Tao and Ziegler concerning $\mathbb{F}_p^\omega$-actions. This result is the first instance of a structure theorem for the universal characteristic factors associated with a non-finitely generated group of unbounded torsion. As an application we derive an alternative proof for the $L^2$-convergence of multiple ergodic averages associated with $k$-term arithmetic progressions in $G$ and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system.