The distribution of directions in an affine lattice: two-point correlations and mixed moments

TL;DR

This paper investigates the statistical distribution of directions in affine lattices, showing that the two-point correlations are Poissonian and establishing the convergence of mixed moments under certain conditions.

Contribution

It proves the existence of Poissonian two-point correlation functions for affine lattice directions and extends results to mixed moments, answering a recent open question.

Findings

Two-point correlation functions are Poissonian.

The limit of mixed moments exists under Diophantine conditions.

The gap distribution differs from the Poisson process, exhibiting a heavy tail.

Abstract

We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most $T$. Str\"ombergsson and the second author proved in [Annals of Math.~173 (2010), 1949--2033] that the distribution of gaps between the lattice directions has a limit as $T$ tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs distinctly from the gap distribution observed in a Poisson process, which is exponential. The present study shows that the limiting two-point correlation function of the projected lattice points exists and is Poissonian. This answers a recent question by Boca, Popa and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments.