2101.01982
ON APPROXIMATION BY RANDOM LÜROTH EXPANSIONS
Charlene Kalle, Marta Maggioni
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 that the almost-sure time average of the approximation coefficients for the random Lüroth map equals Mp = p(2ζ(2)−3)/2 + (2−ζ(2))/2 and spans [MA, ML], by using Birkhoff to deduce uniform distribution of iterates and then computing the CDF via known FL and FA before integrating. The model gives a different but equivalent proof: it identifies θn as an L-orbit observable, applies Birkhoff directly to θ1, and computes the space average by conditioning on branches and using linearity to get a mean of 1/2. Both routes yield the same formula and range; assumptions (invariance/ergodicity of mp×λ, c=0 case) match the paper’s setup.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper introduces a clean randomization of Lüroth maps and establishes robust approximation results. The theorem on approximation coefficients is correct and clearly positioned with respect to classical GLS results. A few steps (e.g., the ergodicity claim and the passage from stationarity to time averages for θn) could be stated more explicitly, but the arguments are standard. The contribution is solid and of interest to specialists in metric number theory and random dynamical systems.