Scaled packing pressures on subsets for amenable group actions

Zubiao Xiao, Hongwei Jia, Zhengyu Yin

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves PP(X_μ,{F_n},f,b)=h_μ(X)+∫f dμ under Condition 2.3, a tempered Følner sequence, μ ergodic, and |F_n|/b(|F_n|)→1 (Theorem 5.1), via a Billingsley-type relation and a variational framework that control all centers, not just a.e. points. The candidate’s lower bound is fine, but the upper bound restricts packings to a large-measure subset Y where Brin–Katok and Birkhoff hold uniformly; it then attempts to pass to X_μ by a countable cover that does not actually cover all of X_μ and gives no control on the complement. Because PP is topological and not absolutely continuous in μ, ignoring a μ-null residual can invalidate the claimed upper bound. Hence the model’s proof is incomplete, while the paper’s argument is correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript gives a complete and rigorous proof of the main identity for scaled packing pressure on generic sets under amenable actions. It integrates a Billingsley-type result and a variational principle with a robust upper bound that does not rely on almost-everywhere uniformity. While largely clear, a few proof roadmaps and reminders of assumptions would further improve readability.