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arXiv · 2308.15275

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

Nihar Gargava·Vlad Serban·Maryna Viazovska
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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.

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Nihar Gargava, Vlad Serban, Maryna Viazovska. 2023-08-29. Moments of the number of points in a bounded set for number field lattices. https://arxiv.org/abs/2308.15275

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