Rough Weierstrass Functions and Dynamical Systems: The Smoothness of the SBR Measure

The paper’s main theorem (Theorem 6.1) proves absolute continuity with an L2 density for the SBR measure under the hypothesis that (ξ,η,y) ↦ Σ_{x∈f^{-1}_{ξ,η}(y)} 1/|f'_{ξ,η}(x)| is bounded and continuous at y=0, via a telescoping identity for ρ, a coarea-type density formula for ρ̂, and a Fourier/Plancherel-type argument that reduces the L2 question to the behavior at y=0 . The candidate solution reaches the same conclusion but by a different route: smoothing in the y-direction, an L2-energy identity, and the 1D coarea formula to uniformly bound the L2 norms of the mollified densities, followed by weak compactness and identification of the limit. Aside from a slightly stronger boundedness formulation (global in y) and a minor coarea-formula convention at critical points, the model’s argument is sound and logically coherent, and it matches the paper’s result under the same transversality-driven hypothesis (verified in the paper for κ ≤ κ0) .