Metric mean dimension, Hölder regularity and Assouad spectrum

The paper proves Theorem A: if (X,d) has finite Assouad dimension and T is α–Hölder, then mdimM(X,d,T) ≤ (1−α) dimA^α(X,d), and if dimA(X,d)>0 then sup{α: T is α–Hölder} ≤ 1 − mdimM(X,d,T)/dimA(X,d). This is stated and proved via a subshift-of-compact-type/spectral-radius approach that bounds the ε-entropy using a transition matrix and the Assouad spectrum (see Theorem A, definitions of mdimM and the Assouad spectrum, and the proof in Section 5: the row-sum bound r(A) ≤ max_i #(j: a_ij=1) together with an ε–cover constructed from ε-separated points; the argument culminates in mdimM ≤ (1−α) dimA^α) . The candidate model gives a different, direct cover-refinement proof: normalize the Hölder constant by scaling, use the Assouad spectrum local covering bound sup_x S(B_{ε^α}(x),d,ε) ≲ ε^{−(1−α)s} for s>dimA^α, and build a d_n–cover by iteratively pulling back ε-covers of T^{j+1}(U) inside B_{ε^α}, which multiplies the number of sets by at most that Assouad-spectrum factor per step; this yields h_ε ≤ log M_s(ε) and hence mdimM ≤ (1−α)s, then s↓dimA^α. Finally, dimA^α ≤ dimA gives the second inequality, matching the paper’s corollary via inequality (11) . The only minor omission in the model writeup is that it does not explicitly state the paper’s standing hypothesis of finite Assouad dimension (used implicitly to ensure dimA^α < ∞), but otherwise the logic is sound and reaches the same bounds by a different method.