Central limit theorems for lattice point counting on tessellated domains

TL;DR

This paper proves that the normalized discrepancy in lattice point counting on tessellated domains, including affine and congruence cases, follows a non-degenerate central limit theorem, using cumulants and mixing estimates.

Contribution

It extends central limit theorems for lattice point counting to tessellated domains and affine and congruence settings, providing new probabilistic results.

Findings

Normalized discrepancy functions satisfy a non-degenerate CLT

Results apply to tessellated, affine, and congruence lattice counting

Uses cumulants and mixing estimates in proofs

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.