Generation of measures by statistics of rotations along sets of integers

The candidate solution proves the two parts of Theorem 10.3—realizing a bounded Radon–Nikodym derivative ρ by a good subset S ⊂ R with MR(S)=1/‖ρ‖∞ and showing absolute continuity of μS,β with an L∞ bound—using a deterministic block-and-threshold (layer–cake) construction and mixed averages A_R(N)[ψ(sα)e(sβ)]. This matches the paper’s statements (Theorem 10.3) and conclusions (MR(S)=1/‖ρ‖∞ and ‖dμS,β/dμR,β‖∞ ≤ 1/‖S‖1,R) exactly . The paper’s proof goes via representation by good weights and a random selection step that requires sublacunarity (Proposition 29.2) , whereas the model’s construction avoids randomness and does not use sublacunarity. The absolute continuity inequality is proved in the paper by a subsequence and comparison argument (eqs. (33.2)–(33.3)) , which coincides with the model’s argument. The only gap in the model’s write-up is a missing justification that, for each fixed β, the limits of the mixed averages with discontinuous cutoffs 1_{A_u}(sα)e(sβ) exist; this follows from continuity on C(T), the boundedness of the linear functionals, and approximation using the total variation measures (which converge to μR,α) chosen so that μR,α(∂A_u)=0 (a standard portmanteau-style argument). With that detail clarified, both are correct, but they use different proof strategies.