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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