Host–Kra factors for ⊕ p∈Pℤ∕pℤ actions and finite-dimensional nilpotent systems

Let $\mathcal{𝒫}$ be a countable multiset of primes and let $G={\oplus <!--FUNCTION\; APPLICATION-->}_{p\in P}\mathbb{Z}∕p\mathbb{Z}$. 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. As an application of our structure theorem 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. Our results provide a counterpart of the structure theorem of Host and Kra (2005) and Ziegler (2007) concerning $\mathbb{Z}$-actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning ${\mathbb{𝔽}}_p^{\omega }$-actions. This is also the first instance of studying the Host–Kra factors of nonfinitely generated groups of unbounded torsion.