FELDMAN-KATOK METRIC MEAN DIMENSION

The paper proves two main claims: (i) a variational principle for FK metric mean dimension via the local FK-entropy function, and (ii) under weak tame growth and bounded mistake density (F(ε)/ε finite), the equality mdim_M = mdim_FK = mdim_M(g). For (i), the paper establishes a fixed-scale identity RFK(T,X,d,ε) = sup_x h_FK(x,ε) using a union lemma (Lemma 2.3) and a nested closed-ball construction, which implies the stated limsup/liminf variational identities after dividing by log(1/ε) (proof of Theorem 1.1) . For (ii), the paper proves mdim_M ≤ mdim_FK via an open-cover/Lebesgue-number argument with a precise combinatorial count of order-preserving matchings, and then shows mdim_M ≤ mdim_M(g) with a parallel argument for mistake balls; both directions use the weak tame growth hypothesis to make the overhead terms vanish in the log(1/ε) scale (Theorem 1.2) . The candidate solution reaches the same conclusions: it proves the fixed-scale identity by a finite-subcover/union argument, and derives the metric-equality results via direct covering lemmas for FK-balls and mistake balls by Bowen balls with combinatorial overhead that vanishes under weak tame growth and F(ε)=O(ε). The approaches are slightly different (direct ball-covering vs. open-cover route), but logically consistent with the paper, and the assumptions match the paper’s (weak tame growth and existence of F(ε)) .