Distribution of Primitve Lattice Points in Large Dimensions

We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set Sd Rᵈ as d goes to infinity, under certain volume conditions on Sd. Our main technique involves exploring higher moment formulas for the primitive Siegel transform. We first demonstrate that if the volume of Sd remains fixed for all d N, then the distribution of the half the number of primitive lattice points in Sd converges, in distribution, to the Poisson distribution of mean 1 2. Furthermore, if the volume of Sd goes to infinity subexponentially as d approaches infinity, the normalized distribution of the half the number of primitive lattice points in Sd converges, in distribution, to the Normal distribution N (0, 1). We also extend these results to the setting of stochastic processes. This work is motivated by the contributions of Rogers (1955), Sodergren (2011) and Strombergsson and Sodergren (2019).