FURSTENBERG SYSTEMS OF HARDY FIELD SEQUENCES AND APPLICATIONS

The paper proves that for Hardy-field a(t) of at most polynomial growth, the Furstenberg systems of b(n)=a(n) mod 1 or b(n)=e(a(n)) are modeled by the unipotent cocycle Sd on Td+1 with invariant measure λ×mTd, where λ arises as a limit law of c(n)=a(d)(n)/d! mod 1; it then classifies λ across the four growth regimes and gives the rational-polynomial reduction to residue classes (Theorem 1.1 and Proposition 4.2). This exactly matches the candidate’s outline: the Sd model, the role of the top coordinate (normalized dth difference/derivative), and the regime-by-regime classification (unique Haar in td log t ≺ a(t) ≺ td+1; absolutely continuous at a(t)∼td log t; Dirac measures when td ≺ a(t) ≺ td log t with all δt arising; and the αtd case giving δα/d! and total ergodicity), as well as the decomposition a=p+ε+ã leading to residue classes. The paper’s proofs compute correlations via Taylor expansion and Lemmas 4.3–4.4, while the candidate uses the discrete Taylor–Newton (forward differences) expansion; these are equivalent in this setting and lead to the same correlation identities and Sd-model. Minor implicit assumptions in the candidate (Δd+1a(n)→0; Δd a(n)−a(d)(n)=o(1)) are standard Hardy-field growth facts used in the paper’s own arguments (via Lemma 3.1 and the Taylor remainder) and do not affect correctness. Therefore both are correct and essentially follow the same proof strategy, with only a stylistic difference (discrete differences versus derivatives) in Step 1 of the modeling argument. See Theorem 1.1 for the five cases, including uniqueness/non-uniqueness and the total-ergodicity statement, and Proposition 4.2 with Lemmas 4.3–4.4 for the Sd-model and correlation matching .