Moments of the number of points in a bounded set for number field lattices

TL;DR

This paper studies the distribution of lattice points in a fixed volume for lattices over number fields, showing convergence to a Poisson distribution as the dimension or degree increases, extending classical results to more general lattices.

Contribution

It extends Rogers' classical results on lattice point distributions from integer lattices to lattices over number fields, demonstrating convergence to Poisson distributions under new conditions.

Findings

Moments of lattice point counts converge to Poisson distribution for large dimensions.

Convergence also occurs when increasing the degree of the number field within certain bounds.

Results generalize classical lattice point distribution theorems to algebraic number field lattices.

Abstract

We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $\omega_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/\omega_K$. This extends work of Rogers for $\mathbb{Z}$-lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field $K$ as long as $K$ varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.