# Modulated crystals and almost periodic measures

## TL;DR

This paper re-analyzes modulated crystals and quasicrystals using modern diffraction theory, revealing their stability and structure as a subclass of strongly almost periodic measures, thus providing a unified mathematical framework.

## Contribution

It offers a modern mathematical diffraction analysis of modulated quasicrystals, clarifying their structure and stability properties.

## Key findings

- Modulated quasicrystals form a subclass of strongly almost periodic measures.
- They are stable under almost periodic modulations.
- The analysis provides a coherent view of these structures.

## Abstract

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations.