Moments of the number of points in a bounded set for number field lattices
CoRR(2023)
摘要
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.
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关键词
lattices,moments,number field
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