Gaussian factors, spectra, and P -entropy

The paper states that for any zero-entropy automorphism S there exists a progression family Pj = {j,2j,…,L(j)j} with L(j)→∞ such that hP(S)=0, and a Gaussian automorphism G with completely positive P-entropy for that P; moreover, S and G are disjoint (no nontrivial Markov wreath products). This is explicitly formulated in Theorem 1, together with the definition of P-entropy and the use of a lemma guaranteeing the existence of such P for deterministic (zero-entropy) S, and disjointness deduced from Theorem 4.1 in [8] as cited in the paper . The paper constructs G via a Sidon-type orthogonal operator T with an orthogonality property (⊥P) that forces independence of suitably chosen Gaussian cylinder sets along P, yielding completely positive P-entropy . The candidate solution gives a different, simpler proof: (i) it constructs P directly from the subadditivity defining zero KS entropy to ensure hP(S)=0; (ii) it takes the Gaussian Bernoulli shift G and, using cylinder approximations and independence across well-separated coordinates, shows hP(G,ξ)>0 for every nontrivial ξ; (iii) it invokes the same disjointness principle. These steps match the paper’s statements but use a different G and proof mechanism for positivity (product-independence vs. Sidon orthogonality). The only mild caveat is that the disjointness step in both treatments relies on a theorem from [8], typically stated for ergodic actions; making ergodicity assumptions explicit would improve precision. Overall, both are correct and consistent with each other, though the proofs differ in construction details and spectral emphasis .