A Counterexample on Multiple Convergence Without Commutativity

The paper proves the existence of ergodic, zero-entropy T and S on a Lebesgue probability space and bounded f, g such that the averages (1/N)∑ f(T^n x)g(S^n x) fail to converge in L2, by constructing T as a skew product over a badly-approximable rotation and S via a conjugation that flips fiber coordinates on a sparse set, then showing oscillation of the triple-correlation averages using the discrepancy skew product and the Hopf–Stepanov ratio ergodic theorem. The main theorem is explicitly stated, the discrepancy model and ratio ergodic input are set up, T is shown ergodic with zero entropy, S is constructed by conjugation, and the two subsequences with different limits are established via equations (2.16) and (2.17), completing the proof . The model’s solution follows the same architecture and cites the same ingredients; the only overstatements are cosmetic (e.g., claiming subsequence values near ±1 rather than the paper’s 0 and 1/2, and attributing S’s properties directly to Proposition 2.4 rather than by conjugacy).