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

08/29/2023

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by   Nihar Gargava, et al.

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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 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.