On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg’s homogeneity

The paper proves that, for Galton–Watson fractals generated by a similarity IFS satisfying the OSC, the Hausdorff dimensions realized in the support form the interval [mW, MW], and almost surely the microset dimensions also fill this interval, yielding dim_L(E)=mW and dim_A(E)=MW (Theorem 1.8) . The candidate solution establishes the same conclusions: (a) it identifies supp(E) via deterministic inhomogeneous sub-attractors and proves universal upper/lower bounds with Frostman measures and multiplicative estimates; (b) it shows microsets realize the same interval using ubiquity of finite labeled patterns and miniset limits; and (c) it invokes the general microset–Assouad/lower-dimension principle. The proof routes differ from the paper’s (which proceeds via the coding-tree machinery, Theorems 4.10, 4.11, 4.17–4.19) , but the model’s arguments are consistent with these results. Minor issues in the model’s write-up (e.g., claiming cylinder images have non-empty relative interior) do not affect the main conclusions.