Central limit theorems for lattice point counting on tessellated domains

TL;DR

The paper addresses the lattice point discrepancy problem for tessellated domains $\Omega_T$ under diagonal semigroup actions, establishing non-degenerate central limit theorems for unimodular, affine unimodular, and congruence lattices. It employs the cumulant method and quantitative multiple mixing (BEG), together with Siegel transforms and smooth approximations, to reduce unbounded discrepancies to bounded, tractable objects and verify the Fréchet–Shohat CLT criteria. The authors derive explicit variances: $\sigma_u^2 = 2\left(\frac{2\zeta(m+n-1)}{\zeta(m+n)}-1\right)$ for unimodular lattices, $\sigma_a^2 = 1$ for affine unimodular lattices, and a detailed expression for $\sigma_c^2$ in terms of $\zeta_N$ and related sums for congruence lattices. These results extend central limit-type phenomena to tessellated lattice-counting domains and varied lattice families, with implications for Diophantine approximation and geometry of numbers.

Abstract

Following the approach of Bj$\ddot{\text{o}}$rklund and Gorodnik, we have considered the discrepancy function for lattice point counting on domains that can be nicely tessellated by the action of a diagonal semigroup. We have shown that suitably normalized discrepancy functions for lattice point counting on certain tessellated domains satisfy a non-degenerate central limit theorem. Furthermore, we have also addressed the same problem for affine and congruence lattice point counting, proving analogous non-degenerate central limit theorems for them. The main ingredients of the proofs are the method of cumulants and quantitative multiple mixing estimates.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Proposition 2.5