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

Nihar Gargava, Vlad Serban, Maryna Viazovska

TL;DR

The paper extends Rogers' mean-value framework to lattices over the ring of integers of number fields, proving that the moments of the number of lattice points in an origin-centered ball converge to the moments of a Poisson distribution with mean $V/\\omega_K$ as the lattice rank grows (and, in structured towers, as the degree grows under height lower bounds). The authors develop a Rogers-type integral formula for number-field lattices, relate higher moments to Grassmannian height zeta functions, and isolate the Poisson term via matrices of type $A_m$, while providing explicit exponential-error bounds via Weil heights and the Bogomolov property. They obtain uniform results across sets of number fields, including cyclotomic towers, with zeta-factor controls that ensure exponential decay of non-Poisson terms in the ambient dimension and field degree. The work advances understanding of fine-scale lattice point statistics in arithmetic settings and suggests potential applications to lattice packing, cryptography, and dynamics on homogeneous spaces. The results are robust under general convex bodies and highlight the essential role of height bounds in obtaining Poisson-type limiting behaviour.

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 $ω_K$ the number of roots of unity in $K$, we show that for lattices of large enough dimension the moments of the number of $ω_K$-tuples of lattice points converge to those of a Poisson distribution of mean $V/ω_K$. 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.

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

TL;DR

The paper extends Rogers' mean-value framework to lattices over the ring of integers of number fields, proving that the moments of the number of lattice points in an origin-centered ball converge to the moments of a Poisson distribution with mean as the lattice rank grows (and, in structured towers, as the degree grows under height lower bounds). The authors develop a Rogers-type integral formula for number-field lattices, relate higher moments to Grassmannian height zeta functions, and isolate the Poisson term via matrices of type , while providing explicit exponential-error bounds via Weil heights and the Bogomolov property. They obtain uniform results across sets of number fields, including cyclotomic towers, with zeta-factor controls that ensure exponential decay of non-Poisson terms in the ambient dimension and field degree. The work advances understanding of fine-scale lattice point statistics in arithmetic settings and suggests potential applications to lattice packing, cryptography, and dynamics on homogeneous spaces. The results are robust under general convex bodies and highlight the essential role of height bounds in obtaining Poisson-type limiting behaviour.

Abstract

We examine the moments of the number of lattice points in a fixed ball of volume for lattices in Euclidean space which are modules over the ring of integers of a number field . In particular, denoting by the number of roots of unity in , we show that for lattices of large enough dimension the moments of the number of -tuples of lattice points converge to those of a Poisson distribution of mean . This extends work of Rogers for -lattices. What is more, we show that this convergence can also be achieved by increasing the degree of the number field as long as varies within a set of number fields with uniform lower bounds on the absolute Weil height of non-torsion elements.
Paper Structure (21 sections, 59 theorems, 288 equations, 1 figure)

This paper contains 21 sections, 59 theorems, 288 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of paper and proof
  3. Rogers integration formula for number fields
  4. Convergence of the higher moment formula
  5. Heights of subspaces
  6. Relating the two types of heights
  7. Rational points of bounded height in Grassmannian varieties over number fields
  8. Towards Poisson distribution
  9. Matrices of type A1m
  10. Upper bounds on moments using Weil heights
  11. Mahler measures and the Bogomolov property
  12. Bounds for contributions from projective space
  13. Summing over ideals
  14. General error estimates for A2m-type terms
  15. General moments using the Bogomolov property
  16. ...and 6 more sections

Key Result

Theorem 1

(Rogers, 1956) Let $\Lambda \subseteq \mathbb{R}^{t}$ be a random unit covolume lattice and let $S$ be a centrally symmetric Borel set of volume $V$. Consider the random variable Then, provided the $\mathbb{Z}$-rank $t$ of the lattices satisfies $t\geq \lceil \tfrac{1}{4}n^{2}+3\rceil,$ it follows that the $n$-th moment of the number of non-zero lattice points in $S$ satisfies where is the $n

Figures (1)

  • Figure 1: Intersection of two balls. The base of the dotted line is at a distance of $R\rho$ from the origin. Cutting the intersection along the dotted line gives a ball in one dimension less and has radius $R\sqrt{1-\rho^2}$. We integrate on the parameter $\rho$.

Theorems & Definitions (114)

  • Theorem
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Remark 6
  • Remark 7
  • Lemma 8
  • Remark 9
  • ...and 104 more