On subshifts with low maximal pattern complexity

The uploaded paper proves that pattern Sturmian sequences (defined by p* x(n)=2n for all n) are exactly of two types in the recurrent case—recurrent simple circle rotation codings or nearly simple Toeplitz—and, in the nonrecurrent case, are either nonrecurrent simple circle codings or almost constant. These are Theorems A and B in the PDF and are derived via the MEF analysis and structural results for non‑superlinear maximal pattern complexity (Theorems C and D and subsequent propositions). The candidate solution reproduces this classification, explicitly corrects the 2^n→2n typo, and follows the same proof skeleton (circle/odometer MEF, 1‑hole Toeplitz⇒nearly simple Toeplitz, exclusion of nearly simple Toeplitz in the nonrecurrent orbit closure, and characterization of nonrecurrent circle codings). In particular, the definitions (Definition 2.8), the main classification (Theorems A and B), and the key lemmas and propositions used (e.g., Propositions 4.7, 4.8, 5.2; Theorem 6.1; Lemma 6.5; Proposition 6.6; and Proposition 2.11) align point‑for‑point with the paper’s argument. Therefore, both are correct and the proofs are substantially the same, with the model’s write‑up serving as a concise restatement grounded in the paper’s framework. Citations: definition and setup ; main theorems A–B ; circle×finite MEF reduction and circle coding structure , , ; exclusion of nearly simple Toeplitz in the nonrecurrent case ; almost‑constant alternative and periodic closure obstruction ; nonrecurrent circle coding characterization .