EXTENDED-CYCLE INTEGRALS OF MODULAR FUNCTIONS FOR BADLY APPROXIMABLE NUMBERS

Yuya Murakami

correctmedium confidence

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

Audit review

The paper’s statements and proofs of (i) convergence of v̂al(x), 1̂(x), ε̂x, (ii) the identity 1̂(x)=2 log ε̂x, and (iii) the weighted-average formula (1.7) for val(x) are consistent and carefully justified, notably via Lemma 6.1 and Theorem 5.1. By contrast, the model’s proof outline mishandles a key cocycle/log term (dropping a non-negligible integral that in general scales like path length) and contains a factor-of-two inconsistency when relating the integral of −dz/(z−x) to log c_n; it also omits the paper’s necessary smallness assumption 2^{-n}a_{i,n}→0 used to control error terms. The model’s final conclusions match the paper, but its derivation is flawed and incomplete.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper gives a clear extension of cycle integrals to badly approximable numbers and proves explicit limit formulas under a precise block-structure hypothesis. The core arguments, especially the link 1̂(x)=2 log ε̂x and the block estimate in Lemma 6.1, are sound. Minor improvements in exposition (highlighting the role of 2\^{-n}a\_{i,n}→0 and giving a schematic view of the cocycle manipulations) would further help readers.