Directional Mean Dimension and Continuum-wise Expansive Z^k-Actions

The paper proves Theorem 1.2: every continuum-wise expansive Z^k-action has uniformly bounded (k−1)-dimensional directional mean dimension, with a proof built on two key ingredients: a finite-window consequence of cw-expansiveness (Lemma 4.3) and a boundary-controlled finite closed cover with order O(N^{k−1}) and small mesh in the dynamical metric (Lemma 4.4). These are then combined to control the width-dimension along a codimension-one tube B_1(<v>^⊥)∩[-N,N]^k, yielding the bound after letting N→∞ and ε→0. The steps and constants are stated precisely in the proof of Theorem 1.2, including the crucial expansion of the control region from [-N,N]^k to [-N−m,N+m]^k to apply Lemma 4.3(2) on every interior point u via the translated block u+[−m,m]^k (see Lemma 4.3 and Lemma 4.4, and the Proof of Theorem 1.2). The candidate solution mimics the outline but contains a decisive gap in Step 4: from smallness of T^{m+q}(W) for a single q with m+q in a thickened boundary, it infers smallness of T^m(W) via a finite-window lemma that requires smallness for all q′ in a full cube [−L,L]^k around m. This necessary uniform control is not guaranteed by joining only over an L-thick boundary set; the paper fixes this by working with W_{N+m} whose mesh is small on the larger block [-N−m,N+m]^k, ensuring T^{u+u′}(W) is small for all u′∈[−m,m]^k and allowing Lemma 4.3(2) to be applied at u. Hence the paper’s argument is correct and complete, while the model’s version misses a key uniformity step. See Theorem 1.2 and its proof, including Lemma 4.3 and Lemma 4.4, which supply the precise constants and coverings to make this work .