Interpolating between Hausdorff and box dimension

The uploaded paper’s main theorem (Theorem 5.2.1) states exactly the master equation for s(θ) and the recursive definition of t_ℓ(s) that the candidate derives: Ts(t) = ((s − log M)/log m)·log n + γ I(t), tℓ(s) = Tℓ−1_s(t1(s)), and the zero of 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 gives dim_θ Λ, with the same simplifications at L = 1 and at θ = γ^{−(L−1)} . The paper’s proof strategy (approximate-square covers at a ladder of scales; method of types; a mass distribution/Frostman-type lower bound) matches the model’s blockwise, large-deviation-based cover and tilted-measure lower bound, differing only in the counting tool (method of types vs. Cramér) but yielding the same rate function I(t) and recursion Ts . The notational setup (γ = log n/log m > 1, M non-empty columns with counts N1,…,NM, and L(θ) = 1 + ⌊−log θ/log γ⌋) also agrees . Finally, the qualitative consequences (existence/uniqueness for each θ, analyticity and strict concavity between phase transitions, countably many breakpoints at γ−k) reported by the model align with the paper’s Corollary 5.2.3 .