2104.04408
Decimation Limits of Principal Algebraic Zd-Actions
Elizaveta Arzhakova, Douglas Lind, Klaus Schmidt, Evgeny Verbitskiy
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper establishes uniform convergence of DNf to Df = −Rf* via Mahler’s inequality and a sharp comparison between Rf and the normalized max of coefficient-weighted linear forms (eq. (5.2)), then passes through Legendre duality to identify the limit; the proof is complete and coherent. The model’s argument relies on a key inequality log|ĥ(m)| ≤ −m·u + Rh(u) that misuses Jensen’s inequality (the direction is reversed), so the critical upper bound sup_r(LNf(r)+r·u) ≤ Rf(u) is unjustified, leaving a gap. See Theorem 1.1 and its proof, including (5.2) and the passage to (DNf)*, in the paper .
Referee report (LaTeX)
\textbf{Recommendation:} reject
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The model proof contains a pivotal misuse of Jensen's inequality to upper-bound coefficients by the Ronkin function, which invalidates the step needed to bound sup\_r(LNf(r)+r·u) from above by Rf(u). The paper addresses this with a robust Mahler-based comparison and the decimation-in-Nth-powers structure, yielding a uniform estimate (its equation (5.2)) and a clean Legendre dual conclusion. Without a correct replacement for the flawed step, the model proof does not establish the main theorem.