Distribution of Primitive Lattice Points in Large Dimensions

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).