Hausdorff dimension of shrinking targets on Przytycki-Urbański fractals

The paper states and proves the three-regime dimension formula dim_H R*(z,λ,γ) exactly as in Theorem 2.5, with assumptions matching the regimes (λγ<1/2 with λ∈E, λγ≥1/2 with unique λ-expansion, and almost-every λ in the transversality window with ν_λ-a.e. z) , built on a precise setup of the IFS, the expanding map E, and the coding machinery . The proofs use Shmerkin’s quantitative consequence of exponential separation (Theorem 3.4) , the unique-expansion counting lemma , and transversality/energy arguments for the typical-parameter case (Section 7) . By contrast, the model’s proof hinges on an unsubstantiated “near-disjointness”/bounded-overlap claim for level-n cylinders under exponential separation, which the paper does not assume and which is generally false in overlapping self-similar/affine systems; the paper instead controls multiplicity via measure estimates (e.g., Corollary 3.5) and energy methods (Section 7) . While the model’s final formulas coincide with the paper, key steps (disjointness and necessity of exact tail agreement under E without counting/measure bounds) are unjustified, so the solution is not a correct proof.