On the dynamical asymptotic dimension of a free Zd-action on the Cantor set

The paper proves that any free Zd-action on a Cantor set has dynamical asymptotic dimension at most 3^d − 1, and by a monotonicity lemma, the same bound holds for any of its extensions (Theorem 2.8 and Corollary 2.10). The core construction is an equivariant quasi-tiling and a 3^d-shift covering using vectors with coordinates in {0, ±E}, which yields an open cover with uniformly bounded F-admissible orbit segments (Lemmas 2.2 and 2.5; Proposition 2.6) . The candidate solution reproduces the same scheme: equivariant tilings on the Cantor base, a 3^d family of shifts ensuring coverage, uniform bounds within a tile, and then pullback to the extension via freeness. The only substantive difference is cosmetic (use of R-interiors and a slightly stronger separation constant). One minor slip in the candidate’s remarks is the claim that the optimal bound for d = 1 is 2; in fact, Guentner–Willett–Yu show DAD ≤ 1 for free Z-actions .