HOST–KRA FACTORS FOR ⊕_p∈P Z/ pZ ACTIONS AND FINITE-DIMENSIONAL NILPOTENT SYSTEMS

Let P be a countable multiset of primes and let G = ⊕_p∈P. 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 L² -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 Z-actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning F ω p -actions. This is also the first instance of studying the Host–Kra factors of nonfinitely generated groups of unbounded torsion.