Monochromatic arithmetic progressions in the Fibonacci, Thue–Morse, and Rudin–Shapiro words

The paper proves exact formulas for A(d) in two regimes—g(d) < τ^{-2} and τ^{-2} ≤ g(d) ≤ 1/2—via the rotation model of the Fibonacci word and a careful interval-intersection/step-distance analysis, yielding A(d) = ⌈τ^{-1}/g(d)⌉ (Theorem 4.6) and A(d) = 2⌈(τ^{-1} − g(d))/g(2d)⌉ (Theorem 4.15) . The candidate solution derives exactly the same two formulas by modeling f as a mechanical (Sturmian) coding and analyzing intersections E_ℓ(J,β), including an even–odd splitting that introduces g(2d). Minor differences (choice of slope parameterization) are purely conventional and do not affect the classification. The approaches and key steps (rotation coding, density, small-step nesting, and an even–odd decomposition for the large-step case) are essentially the same.