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

The paper proves that for G = ⊕_{p∈P} Z/pZ and any ergodic system X of order < k+1, there exists m = O_k(1) and an m-extension Y of X that is an inverse limit of finite-dimensional k-step nilpotent homogeneous systems, with precise “moreover” structure via Host–Kra groups (Theorem 2.11) . It also establishes that systems of order < k are inverse limits of finite-dimensional systems with bounded exponent (Theorem 4.3) . The candidate solution reaches the same conclusion via a nilspace-based route, invoking bounded-depth phase polynomials and compact nilspace structure; this is a different argument from the paper’s Host–Kra/Conze–Lesigne approach. The model contains two issues that do not affect the main conclusion: it (i) conflates pulling back along ϕ_m with the paper’s cohomological m-extensions, and (ii) incorrectly asserts the translation group’s connected component is trivial. Neither is relied upon to attain the core theorem. Overall, both establish the same structural result; the paper’s proof is complete and precise, and the model’s proof is largely correct but would need minor corrections and details.