Spherical equidistribution in adelic lattices and applications

TL;DR

This paper investigates spherical equidistribution in adelic lattices and applies these results to analyze the distribution of directions and free path lengths in lattice point sets, combining geometric, probabilistic, and number-theoretic methods.

Contribution

It introduces new equidistribution results for adelic lattices and applies them to the statistical analysis of lattice point directions and free path lengths, including non-rational translates.

Findings

Proves spherical equidistribution for translates of adelic lattices.

Analyzes the distribution of directions in shifted primitive lattice points.

Provides insights into free path lengths in the Boltzmann--Grad limit.

Abstract

In this paper we study spherical equidistribution on the space of (translates of) adelic lattices, which we apply to understand the fine-scale statistics of the directions in the set of shifted primitive lattice points. We also apply our results to the distribution of the free path lengths in the Boltzmann--Grad limit for point sets such as (possibly non-rational) translates of the lattice points all of whose coordinates are squarefree. Besides the equidistribution results for translates of expanding horospheres, a key ingredient is a probabilistic argument which allows us to tackle the technical difficulty of dealing with characteristic functions of compact sets with positive measure and empty interior.