Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.00986 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916310246490112 |
|---|---|
| author | Han, Jiyoung |
| author_facet | Han, Jiyoung |
| contents | We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set $S_d\subset\mathbb R^d$ as $d$ goes to infinity, under certain volume conditions on $S_d$. Our main technique involves exploring higher moment formulas for the primitive Siegel transform. We first demonstrate that if the volume of $S_d$ remains fixed for all $d\in \mathbb N$, then the distribution of the half the number of primitive lattice points in $S_d$ converges, in distribution, to the Poisson distribution of mean $\frac 1 2$. Furthermore, if the volume of $S_d$ goes to infinity subexponentially as $d$ approaches infinity, the normalized distribution of the half the number of primitive lattice points in $S_d$ converges, in distribution, to the normal distribution $\mathcal N(0,1)$. We also extend these results to the setting of stochastic processes. This work is motivated by the contributions of Rogers (1955), Södergren (2011) and Strömbergsson and Södergren (2019). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of Primitive Lattice Points in Large Dimensions Han, Jiyoung Number Theory Probability We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set $S_d\subset\mathbb R^d$ as $d$ goes to infinity, under certain volume conditions on $S_d$. Our main technique involves exploring higher moment formulas for the primitive Siegel transform. We first demonstrate that if the volume of $S_d$ remains fixed for all $d\in \mathbb N$, then the distribution of the half the number of primitive lattice points in $S_d$ converges, in distribution, to the Poisson distribution of mean $\frac 1 2$. Furthermore, if the volume of $S_d$ goes to infinity subexponentially as $d$ approaches infinity, the normalized distribution of the half the number of primitive lattice points in $S_d$ converges, in distribution, to the normal distribution $\mathcal N(0,1)$. We also extend these results to the setting of stochastic processes. This work is motivated by the contributions of Rogers (1955), Södergren (2011) and Strömbergsson and Södergren (2019). |
| title | Distribution of Primitive Lattice Points in Large Dimensions |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2407.00986 |