DICHOTOMY LAWS FOR THE HAUSDORFF MEASURE OF SHRINKING TARGET SETS IN β-DYNAMICAL SYSTEMS

The paper establishes sharp zero–full Hausdorff f-measure dichotomies for two β-dynamical shrinking-target families: (i) the weighted/product-type set W_d(Ψ,h) under integer bases, via the s_n(Ψ,f) covering functional and a new geometric mass-distribution method, and (ii) the multiplicative set W×_d(ψ,h) for general bases, combining a 1D dichotomy for W1(ψ,h) with a slicing argument. The statements and core steps are explicit in Theorem 1.2 and Theorem 1.4, with the covering/minimization that defines s_n(Ψ,f) in the convergence case (Section 3) and a careful divergence construction using separation, local content estimates, and a large-intersection criterion (Section 4) . For the multiplicative problem, the paper proves the 1D dichotomy for general β (Theorem 1.4(1)) and then deploys a slicing lemma to conclude the higher-dimensional statement (Proposition 7.3 and Proposition 7.5) . In contrast, the model’s convergence parts mirror the paper’s coverings and the s_n(Ψ,f) functional accurately, but the divergence arguments have serious gaps. In Theorem A, the model posits a Moran-type construction with “uniform children per parent” and a lower bound H_f^(τ_j)(G_j) ≥ c s_{n_j} ∏ β_i^{n_j} derived from disjointness; this lower bound does not follow from the given upper-cover costs absent a rigorous separation argument at the working scale. The paper resolves precisely this difficulty via a new geometric decomposition with scale parameters ω_n and a measure μ that satisfies µ(B(x,r)) ≲ f(r)/ω_n^d and then a large-intersection principle (Theorem 2.3) to lift to full measure, an ingredient missing from the model’s argument . For Theorem B(d≥2), the model asserts a strong “slicing inequality” H_f(E) ≍ ∫ H_g(E_{x′}) dx′, which is not provided in the paper and is not valid in this generality; the paper instead uses a one-way slicing lemma to conclude divergence from fiberwise Hg = ∞ on positive-measure many slices . Thus, while the model’s conclusions align with the paper’s, crucial steps are unproven or incorrect. The paper’s statements and proofs are consistent and complete within their stated hypotheses.