On arithmetic functions orthogonal to deterministic sequences

The paper’s Theorem A states exactly the equivalence the model proves: a bounded arithmetic function u is orthogonal to every system in C_{Fec} if and only if, for every Furstenberg system κ of u, the coordinate function π0 is orthogonal to L2 of the maximal Fec-factor of (Xu, κ, S) (the “Veech condition”). The statement and its proof strategy are laid out in the introduction and Section 4, with the key joining identity (equation (4)) and the use of the largest characteristic factor DF as background tools. The authors’ proof of (i) ⇒ (ii) uses Hansel models (to handle non-ergodicity) and a lifting lemma (their Proposition 3.1) to realize joinings along subsequences, plus the strong u-MOMO equivalence (Proposition 2.17), culminating in Eκ(π0 | B(Xu)Fec)=0 (see the development around (51)–(53)). The model’s solution proves the same equivalence via a classical joinings/conditional-expectation argument: (ii) ⇒ (i) is shown by passing through Jρ(f)=Eρ(f|Xu) whose range lies in an Fec-factor of (Xu, κ), and (i) ⇒ (ii) is obtained by contraposition using a relatively independent joining over the maximal Fec-factor, Jewett–Krieger to build a strictly ergodic topological model in C_{Fec}, and a realization-of-joinings lemma along a subsequence. This last step is essentially the lifting lemma the paper develops explicitly; once that ingredient is acknowledged, the model’s argument is sound. Hence both are correct but follow different proof routes (Hansel model + lifting lemma vs. Jewett–Krieger + a standard realization lemma). Key pieces in the paper supporting this assessment include the joining identity and largest factor formalism, the definition of the Veech condition, that Fec is characteristic, the strong u-MOMO equivalence, and the detailed proof of Theorem A (including the Hansel model construction and relatively independent extensions).