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.

翻译：本文研究了欧几里得空间中固定体积$V$的球内格点数的矩，这些格点是数域$K$的整数环上的模。特别地，记$\omega_K$为$K$中单位根的数量，我们证明：对于维数足够大的格点，格点的$\omega_K$元组数量的矩收敛于均值为$V/\omega_K$的泊松分布的矩。这推广了Rogers关于$\mathbb{Z}$-格点的工作。此外，我们证明：只要数域$K$在非挠元素的绝对Weil高度具有一致下界的数域集合中变化，通过增加数域$K$的次数亦可实现该收敛性。