THE DIMENSION SPECTRUM OF THE INFINITELY GENERATED APOLLONIAN GASKET

The paper proves that the infinitely generated Apollonian IFS A has full Hausdorff dimension spectrum DS(A) = [0, dimH(J_A)], with a concrete distortion constant KA ≤ 5.900319, sharp two-sided derivative bounds ∥φ′_{k,n}∥∞ ≍ n^{-2}, θ(A)=1/2 from (2.6), and an 18-step bootstrapping chain of overlapping intervals based on a natural ordering and rigorous dimension estimates for subsystems, culminating in Theorem 4.1 DS(A) = [0, dimH(J_A)] (see the system definition (3.1), the bound KA ≤ 5.900319, the derivative bound 0.45/n^2 < ∥φ′_{k,n}∥∞ < 3.821/n^2, θ(A) = 1/2, Proposition 4.1/Corollary 4.3, and Theorem 1.1/4.1) . The model’s solution outlines the same core ingredients (natural order, distortion/derivative control, pressure formalism) and then appeals to the very result established by the paper to assert full spectrum. While the model does not reproduce the computer-assisted bootstrapping or the explicit distortion constants, its sketch matches the paper’s approach at a high level and reaches the same conclusion by citing that result. Hence, both are correct; the model provides a high-level sketch substantially aligned with the paper’s proof rather than an independent proof.