OVERLAPPING SUBSTITUTIONS AND TILINGS

Shigeki Akiyama, Yasushi Nagai, Shu-Qin Zhang

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

Both the paper and the candidate solution prove the frequency theorem by passing from an expanding overlapping substitution ρ to its associated weighted substitution ρ̃, showing that the indicator weighted tiling v of a fixed-point tiling satisfies ρ̃^k(v)=v, and then invoking the Perron–Frobenius frequency proposition to conclude uniform convergence along van Hove sequences. The paper states the key fixed-point identity and derives Theorem 2.8 from Propositions 2.6 and 2.7, while the candidate supplies the standard disjoint-coverage argument establishing ρ̃^k(v)=v. The approaches are essentially the same, with the model filling in a step the paper only sketches.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper establishes a coherent framework for overlapping substitutions via weighted substitutions and proves a general frequency theorem that interprets non-integer substitution matrices in a natural, measurable way. The arguments are largely correct and well-organized, with careful technical work in the appendix. A brief explicit proof that the indicator tiling is fixed under a power of the weighted substitution would close a small gap in exposition. Overall, the contribution is timely and valuable to the tilings community.