Distribution of Primitive Lattice Points in Large Dimensions

TL;DR

This paper studies how primitive lattice points distribute in high-dimensional symmetric sets, showing convergence to Poisson or normal distributions depending on volume growth, using higher moment formulas for the primitive Siegel transform.

Contribution

It provides new asymptotic distribution results for primitive lattice points in large dimensions, extending previous work with higher moment techniques and stochastic process analysis.

Findings

Distribution converges to Poisson when volume is fixed.

Distribution converges to normal when volume grows subexponentially.

Results extend to stochastic processes.

Abstract

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), S\"odergren (2011) and Str\"ombergsson and S\"odergren (2019).