Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

The paper proves the sharp threshold with precise constructions and estimates: Theorem A (existence, with p(q)/f(q)→0 for any superlinear f) and Theorem B (non‑superlinear complexity forces partial rigidity) are stated up front and established via quasi‑staircase rank‑one systems and a detailed combinatorial/mixing analysis . The key complexity upper bound p(q) ≤ q(2+Σ b_n) (Proposition 2.30) underpins the construction and is then tuned in Theorem 3.3 to beat any prescribed superlinear scale after a q↦qf(q) normalization, which reduces to Theorem A when one substitutes f(q)/q for f(q) . Mixing is obtained by proving rank‑one uniform mixing along carefully chosen times combined with quantitative conditions such as a_n b_n^2/h_n→0 (Proposition 4.11 and Theorem 4.12) . Partial rigidity at non‑superlinear complexity is proved via a Rauzy‑graph decomposition and a measure‑theoretic patching lemma (Proposition 5.24), yielding a uniform δX>0 for every ergodic measure on X . By contrast, the model’s Part A relies on a coarse “boundary‑counting” estimate and the classical staircase schedule to claim p(q)=o(f(q)) for arbitrary superlinear f. This directly conflicts with known behavior of staircase codes (which have at least quadratic complexity) and bypasses the paper’s essential quasi‑staircase parameters and mixing constraints. The model also asserts an unwarranted uniformity (“for all n‑cylinders simultaneously”) in the overlap step of Part B that is stronger than what is proved; the paper obtains partial rigidity via selected base sets and times rather than a single t working uniformly for every cylinder. In short, the paper’s results are correct and complete; the model’s construction and complexity bound in Part A are not supported by the needed combinatorics/mixing conditions from the paper.