Sarnak’s Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow

The paper proves Möbius disjointness for systems whose invariant measures have singular spectrum via the Hellinger/affinity method, after showing that every potential spectral measure of μ is absolutely continuous with respect to Lebesgue and that those coming from the dynamical sequence are singular. This uses Definition 2.6 (potential spectral measures), the Coquet–Kamae–Mendès-France/Bellow–Losert affinity bound, and the Sarnak–Veech ‘CSV’ measure for the Möbius flow; see Theorem 3.1, Corollary 2.10 and Theorem 3.21 in the paper. The candidate solution follows the same route and matches the key steps. The only minor discrepancy is the model’s overstatement that the Möbius spectral measures are “indeed, equivalent” to Lebesgue; the paper only needs and states absolute continuity. Overall, the proofs are essentially the same and correct. Key items are explicitly present in the paper (Theorem 3.1, Definition 2.6, the affinity inequality, and CSV properties) .