Parabolic Carpets

The paper proves that for almost quasi-Bernoulli measures on parabolic carpets, the L^q-spectrum equals the pressure root β(q) and dim_B F = dim_P F = β(0), via a δ-stopping antichain, one-rectangle L^q estimates tied to the 1D spectra of the coordinate projections, and an induced uniformly hyperbolic subsystem to control lower bounds through (almost) sub/super-multiplicativity of a singular value partition function. The candidate solution follows the same decomposition, introduces the same stopping-time antichain, derives the same per-rectangle estimate up to ε, and uses the same pressure/inducing mechanism to identify the critical exponent. Minor differences are expository (e.g., not separating the q ≤ 1 and q > 1 cases explicitly, and invoking the 1D spectrum as an assumption rather than citing the paper’s Theorem 4.1). Overall, the logical steps align closely with the paper’s argument and assumptions, and the conclusions match Theorem 6.1 and its corollaries. Key steps and definitions correspond directly to the paper’s S_δ stopping construction, Jensen/separation reduction, Φ_{s,q} partition sums, and the induced subsystem IN/I∞ argument for the lower bound, as stated in Sections 4–6 of the paper (e.g., Theorem 6.1 and Lemma 6.2).