Publication

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

Maryna Viazovska, Nihar Prakash Gargava, Vlad Serban
2023
Report or working paper
Abstract

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

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Related concepts (50)
Algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
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In algebraic number theory, a quadratic field is an algebraic number field of degree two over , the rational numbers. Every such quadratic field is some where is a (uniquely defined) square-free integer different from and . If , the corresponding quadratic field is called a real quadratic field, and, if , it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers.
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