Hausdorff dimension of double base expansions and binary shifts with a hole

The paper’s Theorem 1.1 gives the four-parameter-regime classification and the dimension formulas, including the disjoint/OSC case q0^{-s}+q1^{-s}=1 and, in the overlapping regime, the “kneading-type” identity π_{q0^s,q1^s}(ℓ_{a,b})=π_{q0^s,q1^s}(r_{a,b}) for s, with a,b the quasi-greedy/lazy codings of the overlap endpoints and ℓ_{a,b}, r_{a,b} the extremal admissible sequences; it also proves continuity and the K(q0) threshold where dim=0. These statements appear verbatim in Theorem 1.1 and Theorem 1.2 of the uploaded paper, and the symbolic characterization of U_{q0,q1} via Ω_{a,b} is stated explicitly (up to a countable set) in the text preceding Theorem 1.1. Hence the candidate solution’s conclusions match the paper’s results and are correct. The model’s proof sketch relies on Bowen-type pressure and weighted kneading; the paper proves the same identities via a generalized Moran construction and direct combinatorial arguments, so the proofs are different in technique. Minor issues in the model solution include (i) stating U=π(Ω_{a,b}) without noting the countable exceptional suffixes ending in a or b and (ii) a loose justification for P(1)<0, which in the paper is handled carefully via Lemmas 2.1–2.5 and Proposition 2.6. Overall, the two agree on all substantive points. See Theorem 1.1 and Theorem 1.2 for the four regimes and the π_{q0^s,q1^s}-equations, and the definition of Ω_{a,b}, ℓ_{a,b}, r_{a,b} and the K(q0) threshold in the introduction and Section 2 (e.g., the statements around Theorem 1.1 and 1.2). The continuity of (q0,q1) ↦ dim_H U_{q0,q1} is also directly proved in Theorem 1.5 of the paper.