2208.09091
A Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions
Mumtaz Hussain, Bixuan Li, Nikita Shulga
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves a six-case classification for dim_H F(Φ1,Φ2) using pressure with potentials ψ1, ψ2 and a two-measure Cantor construction, and establishes the emptiness thresholds. The candidate solution reproduces the same classification, uses the same pressure equations P(T,−s log|T'|)=2 s log B1 (for E2) and P(T,−s log|T'| − s log B1 + (1−s) log B2)=0 (for F), and outlines a matching two-measure Frostman lower bound. Apart from a small notational slip (a −2s vs −s exponent in a heuristic sum), the approaches and conclusions align.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper achieves a sharp, exhaustive classification of dim\_H F(Φ1,Φ2) across exponential and double-exponential regimes, extending the state of the art. The techniques—restricted-pressure upper bounds and a two-measure Cantor construction for lower bounds—are solid and clearly connected to the literature. Minor clarifications and expository improvements would help non-experts follow the pressure manipulations and the role of parameter continuity.