2009.01402
BEYOND SUBSTITUTIONS: THE SPECTRAL THEORY OF REGULAR SEQUENCES
Michael Coons, James Evans, Neil Mañibo
incompletehigh confidenceCounterexample detected
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s continuity claim (Theorem 2) is stated early under too few hypotheses, but later sections add the crucial joint spectral radius gap and a non-vanishing condition that make the argument correct; the model correctly identifies that continuity can fail without an additional gap (and gives a valid atomic counterexample), but its own proof sketch omits an essential non-vanishing assumption needed for the ratio limits it uses. net: the paper’s final claims are correct under the later, stronger assumptions; the model’s critique is valid against the early, incomplete statement, yet its argument also misses a needed hypothesis.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
This paper extends spectral and measure-theoretic methods from substitutions to k-regular sequences and establishes existence and continuity of limiting measures under natural spectral gap conditions. The mathematics is solid and connects well with the literature on Mahler functions, dilation equations, and joint spectral radius. A minor issue is that Theorem 2 in the early statement omits key assumptions (joint spectral radius gap and a non-vanishing condition) that are essential later; adding them would prevent confusion and align the statement with the subsequent proofs.