Polynomial ergodic averages of measure-preserving systems acted by Z^d

The paper’s Theorem 1.3 asserts an equivalence: for a Z^d-system and any polynomial array, almost-everywhere pointwise convergence of the multiple polynomial averages for all bounded functions holds if and only if it holds when the functions are measurable with respect to the Pinsker σ-algebra Pµ(Z^d) . The paper then derives Corollary 1.5 claiming a.e. convergence for all bounded functions in every K-system (trivial Pinsker), i.e., in particular for Bernoulli shifts, for arbitrary m and polynomial arrays . This would imply pointwise a.e. convergence of the triple Furstenberg averages in highly mixing systems, a problem widely regarded as open beyond the bilinear case (the paper’s own introduction only records special cases such as bilinear results and distal systems, with no general m≥3 pointwise theorem) . On inspection, the proof contains a concrete error in the key reduction step. In Section 3, to create a martingale-difference–type orthogonality (needed to apply Lemma 3.1), the argument uses Lemma 2.2 together with the identity (p(·))^{-1}A_{g0} = J_{j,n} where J_{j,n} is defined as A_{g0 + p_j(nQ+i)}; this sign is wrong: for the left-invariant order on Z^d and the definition of Ag, one has (p(·))^{-1}A_{g0} = A_{g0 − p(·)}, not A_{g0 + p(·)}. The paper explicitly uses the incorrect plus sign to deduce E(X_{j,n} | J_{j,n}) = 0 and hence pairwise orthogonality via Lemma 3.1, but this identification is invalid as written . Without this orthogonality, the reduction from B to Pµ(Z^d) (the core of Theorem 1.3) fails. The manuscript also restates a Z-case reduction (its Theorem 1.2) in a form that would already imply the aforementioned open m≥3 convergence in K-systems, indicating the same underlying error appears already in its Z reduction claim . By contrast, the model correctly notes that (1) ⇒ (2) is immediate, and that a global implication (2) ⇒ (1) would yield nonconventional pointwise convergence results currently beyond reach; this aligns with the state of results summarized in the paper’s introduction (bilinear polynomial pointwise results only, e.g., Krause–Mirek–Tao) .