Moments of the number of points in a bounded set for number field lattices
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910333314007040 |
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| author | Gargava, Nihar Serban, Vlad Viazovska, Maryna |
| author_facet | Gargava, Nihar Serban, Vlad Viazovska, Maryna |
| contents | 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 $\mathbb{Z}$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15275 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Moments of the number of points in a bounded set for number field lattices Gargava, Nihar Serban, Vlad Viazovska, Maryna Number Theory Information Theory Dynamical Systems 11H06 (Primary) 52C17, 11G50 (Secondary) 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 $\mathbb{Z}$-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. |
| title | Moments of the number of points in a bounded set for number field lattices |
| topic | Number Theory Information Theory Dynamical Systems 11H06 (Primary) 52C17, 11G50 (Secondary) |
| url | https://arxiv.org/abs/2308.15275 |