Higher moment formulae and limiting distributions of lattice points

TL;DR

This paper establishes functional limit theorems for counting lattice points in affine and congruence lattices, utilizing higher moment formulae for Siegel transforms on homogeneous spaces.

Contribution

It introduces new higher moment formulae for Siegel transforms and applies them to derive limit theorems for lattice point counting.

Findings

Proves functional limit theorems for lattice point counts.

Develops higher moment formulae for Siegel transforms.

Provides tools of independent interest for homogeneous space analysis.

Abstract

We prove functional limit theorems for lattice point counting for affine and congruence lattices using the method of moments. Our main tools are higher moment formulae for Siegel transforms on the corresponding homogeneous spaces, which we believe to be of independent interest.