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

2023/08/29 by Nihar Gargava, Gargava, Nihar, Vlad Serban +3 · 2 citations
Mathematics · #Analytic Number Theory Research #Mathematical Dynamics and Fractals #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2308.15275

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 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.

Cited by

• Module lattices and their shortest vectors
• Integral Matrices of Fixed Rank over Number Fields

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