A REMARK ON ”A NON-SINGULAR DYNAMICAL SYSTEM WITHOUT MAXIMAL ERGODIC INEQUALITY” BY E. H. EL ABDALAOUI

The paper is a brief corrective note: it defines the diagonal-orbit system (X^k, B^k, ν, φ) with ν(A) = (1/3)∑_{n∈ℤ}2^{-|n|} µ_Δ(φ^n A), restates the target pointwise problem (Statement 1), and observes that proving pointwise convergence for product-type observables F on (X^k, ν, φ) would suffice to yield Statement 1 (their Statement 2) . The note also explains that a counterexample showing failure of a pointwise ergodic theorem for all L^∞(ν) functions (using 1_A with A = ⋃_n φ^nΔ) does not address the restricted product class actually needed . The candidate solution shows a short equivalence argument: for the φ-invariant set E where the Cesàro limit exists, ν(E) = µ_Δ(E) by definition of ν; on the diagonal this reduces to the classical nonconventional averages, so Statement 2 is equivalent to Statement 1. Since pointwise a.e. convergence is known for k = 1 (Birkhoff) and k = 2 (Bourgain), but remains open for k ≥ 3, the overall claim in full generality is open as of the cutoff. This is consistent with the paper’s stance and clarifies the reduction in a way the note did not spell out.