Some Factors of Nonsingular Bernoulli Shifts

The paper proves a phase transition and ergodic index result for the half-stationary nonsingular Bernoulli shifts ν_c, establishing the existence of a threshold D>1/6 and showing that for c in (D/(k+1), D/k) the ergodic index is k. This is done via a monotonicity/“randomized product” scheme (Theorem 17) and a k-block reduction (Lemma 18), not by computing D explicitly; see the statement of Theorem 3 and its proof sketch, including the reference to Vaes–Wahl for D>1/6, and the product/ergodic-index arguments (Theorem 3, Proposition 20, Lemma 18) . The candidate solution gives a different, sharper approach: using the Hellinger–Kakutani product formula and Aaronson’s Hellinger integral test to compute ρ_n ≍ n^{-c^2/2}, hence D=√2, and then deducing the ergodic index thresholds. This aligns with the paper’s claims (which do not attempt to identify D), and uses standard results (Kakutani, Aaronson, Danilenko). Thus both are correct; the model provides a sharper constant by a different method.