2009.09516
Fourier Nonuniqueness Sets for the Hyperbola and the Perron–Frobenius Operators
Deb Kumar Giri
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the nonuniqueness (infinite-dimensional AC spaces) for rationally shifted lattice-crosses on the hyperbola and on its right branch via Perron–Frobenius operators and Gauss-type maps. The model’s proof reaches the same conclusions by a short containment argument combined with the classical nonuniqueness thresholds for the unshifted lattice-cross (αβ>1 for the full hyperbola; αβ>4 for a single branch), together with the observation that (Z+1/p)⊂(1/p)Z and (2Z+2/q)⊂(2/q)Z. The statements match Theorems 1.7 and 1.13 in the paper, but the proof strategies differ substantially.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper’s main nonuniqueness statements are correct and align with known product-threshold phenomena. The operator-theoretic approach is technically competent and of interest to specialists, but for the specific rational-shift nonuniqueness claims a simple containment argument from the unshifted thresholds already yields the results. Including a brief acknowledgment of this simpler route would sharpen the contribution and guide readers to what is truly novel here (the dual/transfer-operator perspective and associated constructions).