CALCULATING BOX DIMENSION WITH THE METHOD OF TYPES

The paper proves the stated box-dimension formulas rigorously: for Gatzouras–Lalley (GL) sponges with COSC and coordinate ordering, dim_B Λ_d = s_d (Theorem 2.3), and for Barański sponges, dim_B Λ_d = max_σ s_d(σ) (Theorem 2.5). The argument proceeds via approximate cubes and δ-stoppings (equation (3.1)) and a method-of-types counting scheme leading to a variational formula (Proposition 4.1), with the Barański case handled by sorting approximate cubes by σ-orderings (Section 4.2) . The candidate solution correctly states the formulas and uses a multi‑stopping/approximate‑cube strategy, but it makes a critical counting error in the iterative step: for a fixed prefix u at level n, the exact identity is ∑_v M_{n-1}(uv) L_n(uv)^{p_n} = M_{n-1}(u) L_n(u)^{p_n}, which depends on L_n(u); replacing L_n(u)^{p_n} by ρ^{p_n} and then ‘telescoping’ drops the factor L_n(u)^{p_n}, which can vary with u by powers of ρ. This undermines the claimed telescoping product and the derived covering count. The paper’s method-of-types avoids this pitfall by controlling type classes and Lyapunov exponents, yielding the stated formulas rigorously .