Conditions on the continuity of the Hausdorff measure

The paper proves that for the linear IFS generated by the partition points b_k (with inverse branches g_k mapping [0,1] to [b_k,b_{k-1}]), if (1−h_n) ln n → 0 and sup_k (b_k−b_{k+1})/b_{k+1} < ∞, then H^{h_n}(J_n) → 1; it sets up OSC, the dimension equation ∑(b_k−b_{k+1})^{h_n}=1, an upper bound H^{h_n}(J_n) ≤ 1 via cylinder covers, and a delicate lower bound via density theorems and Lemma 4.1 (which requires condition (2)), culminating in Theorem 4.8. These steps are explicitly laid out and coherent in the manuscript . The model presents a shorter approach based on a self-similar (h_n-conformal) measure and a Frostman-type estimate, and claims condition (2) is unnecessary. However, its key estimate incorrectly bounds the number of (m−1)-level “parents” intersecting a given interval by n+2; there can be arbitrarily many tiny parents inside the interval. This gap prevents the claimed uniform Frostman inequality as written. The argument can be repaired by replacing the faulty counting step with the inequality a^h ≤ δ^{h−1} a for a ≤ δ (taking δ = |I|/n), yielding μ_n(I) ≤ n^{1−h_n}(1+2/n)|I|^{h_n} and the same limit under (1). But because this fix is not in the submitted solution, the model’s proof is formally flawed. The paper’s proof, using both (1) and (2), is correct as written.