Krieger’s type of nonsingular Poisson suspensions and IDPFT systems

The paper proves that for every infinite countable amenable group Γ there are sharply weak mixing nonsingular Poisson Γ-actions of each Krieger type II∞ and IIIλ for λ∈[0,1] (Theorem 0.2) . Its construction uses a specific Rokhlin-tower-based base action inside Aut1, quantitative Aut1/L1 criteria (Lemma 1.4) and an IDPFT decomposition of the Poisson suspension, which, once conservative by Lemma 1.3, yields sharp weak mixing via Corollary 1.5 (Fact 1.2) . The type computations match the paper’s sections: II∞ via a restricted/MH-product invariant measure (Theorem 2.5 and ensuing argument) ; III0 with r(Γ*)={1} but no equivalent invariant measure (Theorem 3.1) ; IIIλ using a Skellam-based essential value argument to get r(Γ*)={λn} (Theorem 4.1 and footnote) ; and III1 by making the multiplicative subgroup dense (Theorem 5.1) . The candidate solution outlines the same pipeline and relies on the same mechanisms. Minor issues: it mislabels the conservativeness tools as “Lemmas 1.5–1.6” (the paper uses Lemma 1.3 plus Lemma 1.4) and introduces an unnecessary “circle-phase rotation” description not used in the paper. These are presentation inaccuracies, not mathematical errors. Hence both are correct and essentially the same proof.