QUANTITATIVE WEAK MIXING FOR RANDOM SUBSTITUTION TILINGS

The paper proves that for μ-a.e. directive sequence x, and for almost every deformation parameter 𝔡 in the moduli Mx (invertible Ruelle–Sullivan), twisted ergodic integrals over cubes satisfy |S^x_R(f,λ)| ≤ C R^{d−α_μ+ε} uniformly for 0<||λ||≤B, yielding d_f^−(λ) ≥ 2α_μ. This is Theorem 1 and its mechanism via a quantitative Veech criterion (Proposition 4/Proposition 6), plus an Erdős–Kahane exclusion for deformations (Section 10), and the kernel/L^2-to-measure step (Lemma 1) . The candidate solution follows the same strategy: S-adic renormalization to supertile scales; Oseledets control on H^1 with dim E^+>d; a quantitative non-resonance criterion along return times; Erdős–Kahane parameter exclusion for 𝔡; and the kernel estimate to deduce local dimension. Minor differences are expository (e.g., the candidate’s heuristic balance of growth exponents and brief handling of boundary terms), but there is no substantive conflict with the paper’s logic.