Multifractal Level Sets and Metric Mean Dimension with Potential

The paper proves the claimed conditional variational principles for the upper/lower Bowen and packing metric mean dimensions with potential on the multifractal level sets K_α under specification (Theorem 1.1) by establishing the chain of inequalities mdim_B ≤ Λ ≤ H ≤ mdim_B (together with the general bound mdim_B ≤ midm_P), from which the equalities follow for both upper and lower versions. The key steps are: (i) upper bounds mdim_B ≤ Λ and mdim_P ≤ Λ (Propositions 3.1 and 3.2), obtained by compressing coverings/packings of K_α into coverings by sets P(α,δ,n) with uniform control; (ii) Λ ≤ H, via spanning/separated sums controlled by scale-ε measure entropy and a small-oscillation argument (Proposition 3.3); (iii) H ≤ mdim_B, via a Moran-like fractal construction and a generalized pressure distribution principle (Proposition 3.6). See the paper for precise definitions and steps, including the auxiliary constructs Γ_ψ^ϕ(α,ε) and Λ^ϕ_mdim^M(f,K_α,d,ψ) and the separated–spanning comparison (Proposition 2.5) used repeatedly. Theorem statement and definitions appear in the introduction and Section 2, with proofs in Section 3. By contrast, the candidate solution reverses a key fixed-scale inequality and asserts a stronger fixed-ε variational identity that the paper does not claim and does not generally hold without additional care. Specifically: (1) it claims Λ(ε) ≤ packing-quantity P_ε, whereas the paper proves the opposite fixed-scale direction needed for the upper bound, i.e., P_ε ≤ Γ_ψ^ϕ(α,ε), leading to mdim_P ≤ Λ^ϕ_mdim^M after passing to scales (Proposition 3.2). (2) it asserts an exact fixed-ε equality Λ_ψ^ϕ(α,ε) = sup_{∫ϕ dμ=α}(h_μ(f,ε) + log(1/ε)∫ψ dμ). The paper instead establishes only Λ ≤ H at fixed scales via δ-level entropies and a loss factor (e.g., log(1/(4ε))) in the measure-theoretic lower bounds (Lemma 3.5) and then recovers equality only after dividing by log(1/ε) and taking limsup/liminf; see the measure-entropy bounds and the pressure distribution argument. The paper’s lower bound further uses a Moran-like fractal and a generalized pressure distribution principle (Lemma 3.22), which the model’s outline does not address.