Diffraction intensities of a class of binary Pisot substitutions via exponential sums

TL;DR

This paper investigates diffraction intensities of binary Pisot substitutions using exponential sums, proving that intensities are confined within the Fourier module, and extends results to random substitutions.

Contribution

It provides a new constructive proof that diffraction intensities are limited to the Fourier module for a class of binary Pisot substitutions, with applications to random substitutions.

Findings

No diffraction intensities outside the Fourier module

Constructive proof using algebraic integers

Application to random substitution models

Abstract

This paper is concerned with the study of diffraction intensities of a relevant class of binary Pisot substitutions via exponential sums. Arithmetic properties of algebraic integers are used to give a new and constructive proof of the fact that there are no diffraction intensities outside the Fourier module of the underlying cut and project schemes. The results are then applied in the context of random substitutions.