2508.19743
Superoptimal continued fractions
Slade Sanderson
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:57 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The candidate solution reproduces the paper’s Theorem 2.9 essentially verbatim: it proves the algebraic identity Θ(x,p_n/q_n)=g(G^{n+1}(x,0)) (paper’s Proposition 2.8) and then uses hitting times of the induced system together with equidistribution of the orbit (x,0) to derive (i) Θ ≤ ε along the selected convergents and (ii) lim inf n(k)/k ≥ 1/ν̄_G(Δ) ≥ C. This matches the paper’s construction and proof, differing only by an index shift and terminology. Key steps (the identity, the induced subsequence of convergents, and the equidistribution statement for (x,0)) align exactly with the paper’s statements and proofs.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript cleanly synthesizes natural extension methods with Seidel contraction to produce families of continued fraction algorithms achieving prescribed uniform approximation quality and convergence density. The main theorem is proved succinctly using a classical identity and equidistribution of orbits from the vertical fiber (x,0). The presentation is clear and well-organized, with illustrative examples. Minor edits to highlight key assumptions and provide brief derivations would improve accessibility.