Absence of Absolutely Continuous Diffraction Spectrum for Certain One-Dimensional S-adic Tilings

Statement A in the candidate solution reproduces the paper’s sufficient criterion for absence of absolutely continuous diffraction and follows the same renormalisation inequality on the Radon–Nikodym matrices H(k), culminating in the Lyapunov-type gap (22) implying H(1)=0 (cf. Proposition 3.12 and Theorem 3.14 in the paper) . The paper also supplies the uniform trace bound needed for the integration step (Lemma 3.13) . By contrast, for Statement B the candidate’s proof crucially assumes the existence of a measurable set G and a uniform contraction c<1 for a normalised block cocycle, derived solely from a.e.-nonsingularity; they acknowledge this step is not justified under the stated hypotheses. The paper avoids this gap entirely: in the binary constant-length setting (Setting 3.17), it constructs a compact-domain cocycle C over an ergodic skew-product R on T×Y, applies Oseledets/Furstenberg–Kesten to obtain Lyapunov exponents, and then derives a strict inequality using Mahler measure estimates (Lemma 3.26 and Lemma 3.27), which yields the desired gap (22) almost surely and thus no absolutely continuous diffraction (Theorem 3.28) . Therefore, the paper’s arguments are complete and correct, while the model’s Statement B is incomplete without an additional non-degeneracy assumption it proposes ad hoc.