MINIMAL AMENABLE SUBSHIFT WITH FULL MEAN DIMENSION

Zhengyu Yin, Zubiao Xiao

correcthigh confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that for any infinite countable amenable group G and polyhedron P there exists a minimal subshift X ⊂ P^G with mdim(X,G) = dim(P), via a primely-congruent tower of exact tilings, a forcing/marker scheme to ensure minimality, and a lower-bound principle for mean dimension based on a positive-density set of "free" coordinates. This is Theorem 1.2 in the uploaded preprint and its proof in Sections 3.2–3.4 (tilings: Lemma 2.6; construction and minimality: Lemma 3.5; lower bound: Proposition 2.13/Corollary 2.14; conclusion: Theorem 3.7) . The candidate solution follows essentially the same construction: a nested, primely-congruent tiling tower; finite block families that force syndetic occurrences of prior-level blocks to achieve minimality; and a density-one set of coordinates carrying [0,1]^m-freedom to obtain the lower bound, with the standard upper bound giving equality. Differences are only in presentation (e.g., using trimmed interiors versus “large-diameter subintervals”), not in substance.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript successfully extends the full mean-dimension minimal subshift construction from Z-actions to general countable amenable groups using a primely-congruent exact tiling tower and a standard forcing scheme. The minimality argument and the metric lower bound for mean dimension are executed correctly, answering an open question from earlier work. Minor editorial enhancements would improve readability and reproducibility of the quantitative steps.