Distribution of Primitive Lattice Points in Large Dimensions

2024/07/01 by Jiyoung Han, Han, Jiyoung · 1 citation
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2407.00986

Abstract

We investigate the asymptotic behavior of the distribution of primitive lattice points in a symmetric Borel set S_d⊂\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∈ \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).

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