DECAY OF FOURIER COEFFICIENTS FOR FURSTENBERG MEASURES

The paper’s Theorem A proves that for non-elementary µ on SL2(R) with finite second moment, the Furstenberg measure ν is Rajchman, i.e., ν̂(k) → 0, and does so via a renewal-operator method adapted to low-moment regimes (Sections 3–5) . The model’s outline hinges on an unsubstantiated Doeblin–Fortet-type contraction on Hölder seminorms derived from κ(α) = ∫|g′(x)|^α dµ(g)dν(x) < 1, and then implicitly factorizes expectations along products of random matrices. This step is invalid in general (the required factorization fails, and such Hölder-contractive framework is precisely unavailable at finite second moment per the paper’s analysis), so the proposed proof is flawed, even though the final claim matches the paper’s result.