Intermediate dimensions of Bedford–McMullen carpets with applications to Lipschitz equivalence

The uploaded paper proves an explicit, complete formula for the intermediate dimensions of any Bedford–McMullen carpet for all θ∈(0,1], identifying dim_θ Λ as the unique s solving γ^L θ log N − (γ^L θ − 1) t_L(s) + γ(1 − γ^{L−1} θ)(log M − I(t_L(s))) − s log n = 0, with t_{ℓ+1}(s) defined by the recursion T_s(t)=((s−log M)/log m)·log n+γ I(t) and t_ℓ(s)=T_s^{ℓ−1}(((s−log M)/log m)·log n) (Theorem 2.1). The paper also establishes existence, uniqueness, the necessary multi-scale cover, and the matching lower bound via a tailored mass distribution principle, fully resolving earlier questions. These statements appear verbatim in the results section and proofs (including special cases L=1 and θ=γ^{-L}), and the general theory of Ss_{δ,θ} and the mass distribution principle is explicitly stated and used. In contrast, the model claims the equality dim_θ Λ = s(θ) was likely open as of the cutoff; this is contradicted by the paper’s main theorem and proof. See Theorem 2.1 and its surrounding discussion for the exact formula and proof outline, including the recursion (2.2)–(2.4), and the proof machinery (method of types; multi-scale cover; intermediate-dimension mass distribution principle).