ON THE HOMOGENEOUS ERGODIC BILINEAR AVERAGES WITH 1-BOUNDED MULTIPLICATIVE WEIGHTS

The paper states the desired bilinear Möbius-disjointness theorem (Theorem 5.1) and outlines a Bourgain-style oscillation/transference proof, but key steps are only sketched or deferred. In particular: (i) the proof shifts to averages of the form f(T^n x)g(T^{-n} x) without explicitly addressing non-invertible T or giving a reduction from general a,b to the symmetric (1,−1) case; (ii) the Wiener–Wintner DKBSZ criterion (Proposition 3.1) is stated as a “straightforward generalization” with no proof; and (iii) the “fundamental lemma” (Lemma 5.4) is cited as a compilation of Bourgain’s results with details promised in a revised version. These omissions leave the argument incomplete, even though the overall strategy is plausible and consistent with known techniques . By contrast, the candidate solution gives a complete, standard proof: reduce to bounded functions; decompose via the Z2 (Conze–Lesigne/Host–Kra) factor; use the uniform Wiener–Wintner double recurrence to kill Z2-orthogonal parts; and invoke Frantzikinakis–Host orthogonality of aperiodic multiplicative functions to nilsequences for the Z2-structured part. This route is rigorous and widely accepted, yielding the desired limit. Therefore: paper incomplete, model correct.