ON THE PACKING DIMENSION OF WEIGHTED SINGULAR MATRICES ON FRACTALS

The candidate solution reproduces the paper’s method: the weighted Dani correspondence, wedge representations and budgets w_l, construction of height functions f_{ε,η̂} with averaged contraction, and a covering/packing argument whose normalization yields the 1/(a1+b1) factor. It also matches the paper’s reduction from Sing(a,b,ω) to a φ1-threshold via Lemma 7.2 and then applies the general theorem (Theorem 1.6) to get the ω-dependent drop. The explicit η_l for K = [0,1]^{m n} also coincide with the paper’s values. The only gaps are expository: the model uses an exponential reparameterization g_t = diag(e^{a_i t}, e^{-b_j t}) instead of the paper’s power-law g_t = diag(t^{a_i}, t^{-b_j}), and it implicitly assumes all hypotheses on K (equal contraction ratio, OSC, and dim_H(K_{ij})>0) without restating them. Mathematically, the approach and bounds agree with the statements and proofs in Theorem 1.3 and Theorem 1.6 of the paper, including the definitions of w_l and the ω-term η_1 a_m b_n ω/(a_m + b_n + a_m ω) and the 1/(a1+b1) normalization. These are explicitly present in the paper’s statements and lemmas (Dani correspondence, contraction hypothesis, and the final dimension bounds). See Theorem 1.3 including (3)–(4) and the definition of w_l, Theorem 1.6 and its two bullets, the contraction scheme in Propositions 5.1–5.3, and the abstract dimension drop in Theorem 6.5, as well as Lemmas 7.1–7.2 for the Diophantine-to-dynamics dictionary.