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2402.19151

Approximations of symbolic substitution systems in one dimension

Lior Tenenbaum

correctmedium confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper proves that the IHS Ωn(ω0) converges to the substitution subshift Ω(S) iff every directed path in the defect graph G(S) starting from a 2-word of ω0 has no closed subpath, equivalently that such paths have length < |A|^2 (Theorem 2.1; definition of G(S) in Definition A.3) . It also establishes exponential rates: for good approximants, dH(Ωn(ω0), Ω(S)) ≤ C1 θS^{-n}, and for periodic ω0 a matching lower bound C2 θS^{-n} (Theorem A.5; Section 3 discussion) . The candidate solution reproduces the convergence criterion via the same “defective 2-words follow edges in G(S)” mechanism, and derives the rate using PF growth. For the lower bound, the model invokes Mossé recognizability (a different but standard route), whereas the paper proceeds via complexity bounds (Coven–Hedlund) and a counting argument. Both arguments are logically consistent with the paper’s statements; they differ in technique, not conclusion. The paper’s proofs are given as outlines referencing prior results (e.g., Lemma A.4; Lemma A.6) rather than fully detailed derivations, but the claims align with standard theory, and the model’s solution fills in many operational steps without contradicting the paper’s assumptions .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

A practically useful convergence criterion via a defect graph and optimal exponential rates are presented for one-dimensional primitive substitutions. The exposition is clear, examples are illustrative, and the results have direct implications for spectral approximation. Some technical steps are sketched rather than proved in full; adding short self-contained proofs (or more precise attributions) would improve readability and self-containment.