Attainable forms of intermediate dimensions

The paper proves a necessary-and-sufficient characterization for the possible lower and upper θ–intermediate dimension profiles of bounded subsets of R^d in terms of: (i) order/continuity constraints and (ii) the sharp Dini-derivative bound D^+h(θ) ≤ ((h(θ)−λ)(α−h(θ)))/((α−λ)θ) when dimL F = λ and dimA F = α, and conversely realizes any admissible pair via Moran constructions (Theorem B with Definition 1.3) . The candidate solution states exactly these two directions: the necessity (monotonicity, continuity on (0,1], dim0 equal to dimH, and the sharp Dini-derivative bound) and the sufficiency via homogeneous/inhomogeneous Moran sets achieving arbitrary admissible h_lower ≤ h_upper with common value at 0. The paper’s derivation of the bound via cover conversion and optimization over scales (δ → δ^β), partitioning by diameter ranges and using Assouad/lower dimension growth on local covering numbers (Theorem 2.6 and inequality (2.2)) is the same core method described by the candidate . The realization is also aligned: equal-diameter (“flat”) covering for homogeneous Moran sets (Lemma 2.9) gives a sliding-window formula for upper intermediate dimensions, then bump/connector constructions produce any admissible h; finally, a union trick yields simultaneous prescription of lower and upper profiles (Theorem 3.8, Theorem 3.10, Corollary 3.12) . Minor presentational discrepancies in the candidate (e.g., describing “homogeneous Moran” with variable m_n) do not affect correctness relative to the paper’s results. Overall, both present the same theorem and essentially the same proof strategy.