# Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings

## TL;DR

This paper develops a method using Fourier matrix cocycles to analyze the pure point diffraction spectrum of 1D Pisot inflation tilings, providing explicit formulas for the Fourier transforms of Rauzy fractal windows and eigenfunctions.

## Contribution

It introduces a new approach based on Fourier matrix cocycles to explicitly compute the diffraction spectrum and eigenfunctions for Pisot inflation tilings, including complex Rauzy fractals.

## Key findings

- Derived a transfer matrix equation for diffraction analysis.
- Obtained closed-form Riesz product expressions for Fourier transforms.
- Applied the method to the Tribonacci tiling example.

## Abstract

Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot--Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier matrix cocycle in internal space. This cocycle leads to a transfer matrix equation and thus to a closed expression of matrix Riesz product type for the Fourier transforms of the windows for the covering model sets. In general, these windows are complicated Rauzy fractals and thus difficult to handle. Equivalently, this approach permits a construction of the (always continuously representable) eigenfunctions for the translation dynamical system induced by the inflation rule. We review and further develop the underlying theory, and illustrate it with the family of Pisa substitutions, with special emphasis on the Tribonacci case.