BOX DIMENSIONS OF (×m,×n)-INVARIANT SETS

The paper proves, for any sofic subshift Σ with presentation graph G and irreducible components {Gi}, that dim_B Π(Σ) = max_i { h(Σ_{Gi})/log n + max_{j∈{i}+} h(πΣ_{Gj})(1/log m − 1/log n) } (Theorem 1.2) and gives matching lower/upper bounds via approximate-square scale choices k(δ), l(δ) and entropy counts across components and their forward cones . The candidate solution derives exactly the same formula using the same approximate-squares device, a bridge argument for the lower bound, and an entropy-maximizing component along each prefix for the upper bound. The steps align closely with the paper’s proofs, differing only in presentation; no substantive gaps remain.