# Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres

## TL;DR

This paper establishes Poissonian pair correlation for directions in multi-dimensional affine lattices under Diophantine conditions and introduces escape of mass estimates for horosphere averages, advancing understanding of lattice point distributions.

## Contribution

It proves convergence of moments and Poissonian pair correlation for affine lattice directions in higher dimensions, extending previous distribution results with new escape of mass estimates.

## Key findings

- Pair correlation function is Poissonian in dimension 3 and higher.
- Convergence of moments for directions of affine lattice vectors.
- Escape of mass estimates for horosphere averages in affine lattice space.

## Abstract

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Str\"ombergsson and the second author, and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in the space of affine lattices.