# Aperiodic order and spherical diffraction, II: Translation bounded measures on homogeneous spaces

## TL;DR

This paper develops a general framework for auto-correlation measures of aperiodic point sets and measures in homogeneous spaces, with specific formulas for weighted model sets, advancing diffraction theory in these contexts.

## Contribution

It extends auto-correlation analysis to invariant measures on homogeneous spaces and provides explicit formulas for weighted model sets, facilitating diffraction studies.

## Key findings

- Auto-correlation measures are characterized for invariant random point processes.
- Explicit formulas for auto-correlation of weighted model sets are derived.
- Framework connects auto-correlation with positive-definite distributions in symmetric spaces.

## Abstract

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $\mathbb R^n$. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.