# Diffraction of a model set with complex windows

## TL;DR

This paper analyzes the diffraction pattern of a ternary inflation tiling derived from the plastic number, using Fourier matrix cocycles to compute the diffraction measure of the associated model sets with Rauzy fractal windows.

## Contribution

It introduces a method to explicitly compute the pure point diffraction measure of a complex model set using a Fourier matrix cocycle and a closed-form formula for the Fourier transform of Rauzy fractals.

## Key findings

- Explicit formula for the diffraction measure obtained.
- Rapid convergence of the infinite product for Fourier transform.
- Connection between Rauzy fractals and diffraction patterns.

## Abstract

The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging infinite product.