2504.20718
Lévy-Khintchine Theorems: effective results and central limit theorems
Gaurav Aggarwal, Anish Ghosh
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves an effective law of large numbers and a CLT for counts of best approximations via a non-smooth observable f on SL_{m+n}(R)/SL_{m+n}(Z), using the EMEI hypothesis, smoothing, decay of correlations, and a cumulant method. Its variance is the full Green–Kubo sum over time-lag covariances. The candidate solution matches the dynamical coding and the almost-sure error term, but incorrectly identifies the CLT variance as Var_{μ_X}(F) (zero-lag only), omitting the nonzero covariance contributions that the paper explicitly retains.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper cleanly establishes effective LLN and CLT statements for best approximations in all dimensions under a quantitative multi-equidistribution (EMEI) hypothesis, using a carefully constructed non-smooth observable and standard smoothing, mixing, and cumulant tools. The dependence of the constants solely on (m,n, norms) is convincing. Minor clarifications (boundedness of f, brief discussion of the Green–Kubo variance, and pointers to technical lemmas) would further improve readability.