AMENABLE GROUP ACTIONS ON Lp LATTICES

Antonio M. Scielzo

correctmedium confidence

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

Audit review

The paper’s decomposition theorem (Theorem 2.16) is proved via approximate multicovers and a careful tower selection that guarantees disjointness across shapes, near-full coverage, and a quantitative boundary bound. The candidate’s MIS-based construction contains critical flaws: (i) it cannot justify X = ⋃_{g∈E0} gW from an E1-maximal independent set; (ii) it makes all shape-towers share at least the g=1 block, so different-shape towers are not disjoint; and (iii) its boundary accounting drops multiplicities and imposes an unjustified smallness condition δ < ε/S_T depending on the tiling family. The paper’s argument is consistent and complete for the stated hypotheses, whereas the model’s proof fails on key logical steps and assumptions.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The main decomposition theorem for non-singular actions is proved with a well-structured and correct adaptation of OW techniques, strengthened by DHZ tilings for arbitrary amenable groups. The argument is technically sound and broadly useful, with applications to the model theory of G–Lp lattices. Minor editorial improvements would aid readability but do not affect correctness.