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

Let P be a countable multiset of primes and let G = p∈P Z/pZ.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 (Definition 2.5) of an inverse limit of finite dimensional k-step nilpotent homogeneous spaces (Theorem 2.12).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 (Theorem 2.13) and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system (Theorem 2.14).Our results provide a counterpart of the structure theorem of Host and Kra [31] and Ziegler [54] concerning Z-actions and generalizes the results of Bergelson Tao and Ziegler [5], [6] concerning F ω p -actions.This is also the first instance of studying the Host-Kra factors of non-finitely generated groups of unbounded torsion.Conventions: We use X to denote a probability space.For technical reasons we assume that any probability space X = (X, B, µ) is regular 1 and separable modulo null sets.We let (U, •) denote a compact abelian group and we assume that all topological