ON d̄-APPROACHABILITY, ENTROPY DENSITY AND B-FREE SHIFTS

The paper’s Theorem 6 establishes the equivalence among (1) a d̄H-limit of a decreasing sequence of mixing sofic shifts, (2) σ-surjectivity plus the d̄-shadowing property, and (3) chain mixing plus d̄-approachability, with the proof routed through standard lemmas: mixing sofic ⇒ d̄-shadowing (Lemma 7), passage of d̄-shadowing to limits (Lemma 8), d̄-shadowing ⇒ d̄-approachability (Lemma 10), and d̄-shadowing + σ-surjectivity ⇒ chain mixing (Lemma 9), together with the observation that Markov approximations of a chain-mixing shift are mixing sofic; see Theorem 6 and surrounding lemmas in the paper . The candidate solution proves the same cycle (1 ⇒ 2 ⇒ 3 ⇒ 1) using essentially the same ingredients: a uniform connector length for mixing sofic shifts (equivalent to the specification-with-fixed-gap formulation used in Lemma 7), a direct concatenation argument for d̄-shadowing, the standard chunking argument for d̄-approachability, and a Rauzy-graph connectivity/aperiodicity argument for chain mixing. These track the paper’s route closely, differing mostly in presentation details. Minor presentational gaps (e.g., ensuring the shadowing is applied to an infinite block sequence and a brief justification that at least some long blocks have low mismatch density) are easily patched and do not affect correctness. Overall, both are correct and substantially the same in proof strategy.