2406.05429
CENTRAL LIMIT THEOREMS FOR LATTICE POINT COUNTING ON TESSELLATED DOMAINS
Sourav Das
correcthigh confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves CLTs for lattice-point counts in tessellated domains on unimodular, affine, and congruence spaces with explicit variances, using a cumulant method plus quantitative multiple mixing and a precise T=2^M tessellation. The model arrives at the same results and variances via a different route (Rogers identities + cumulants), but omits an essential hypothesis u1+...+um=n that the paper states up front and uses repeatedly. Variance constants and the tessellation reduction match the paper’s statements and computations. See theorems and computations of σu^2, σa^2, σc^2, the tessellation Ω2^M=⊔k c0^{-k}Ω2, and volume/log growth in the paper.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The paper cleanly establishes CLTs for lattice point counts in natural tessellated domains across unimodular, affine, and congruence settings, deriving explicit variances and using a robust cumulant+multiplicative-mixing framework. The reduction to T=2\^M and the tessellated structure are leveraged effectively. A few computations in the unimodular and congruence sections are presented succinctly by analogy with the affine case; modest elaboration would enhance readability. Overall the results are correct and of solid interest to the area.