RECURRENT SET ON SOME BEDFORD-MCMULLEN CARPETS

The paper’s Theorem 1 gives exactly the same Hausdorff-dimension formula for the quantitative recurrent set W(K,T,ψ) on Bedford–McMullen carpets with uniform fibres (N_i ≡ N), split into the two regimes log_{m1} m2 ≷ 1+τ1, with τi = liminf ℓ_i(n)/n and ℓ_i(n) = −log_{m_i} ψ(n) (definitions and main statement appear in Section 2; see Theorem 1) . The candidate’s final formula matches Theorem 1 term-by-term, including the boundary continuity at log_{m1} m2 = 1+τ1. For the upper bound, both use approximate-square coverings, though the paper implements it via a “nine-rectangles” decomposition and counting of approximate squares inside each rectangle, yielding the same two competing thresholds and the same case distinction . For the lower bound, the paper constructs a piecewise Bernoulli (product-type) measure and computes lower local dimensions via entropies H1(p), H2(p), then optimizes over p to obtain the claimed bound . The candidate instead sketches two explicit Moran constructions (vertical- and horizontal-dominated) on approximate squares. Thus, results coincide, but the proof strategies differ in the lower-bound step. Overall, both are correct; the paper’s argument is complete modulo standard details indicated (e.g., case analyses and expected-value bounds in Lemmas 4–5 are outlined but consistent) .