# Substitution-based structures with absolutely continuous spectrum

## TL;DR

This paper introduces new substitution-based structures with purely absolutely continuous diffraction and mixed dynamical spectrum, expanding the understanding of spectral properties in aperiodic sequences.

## Contribution

It generalizes Rudin's construction to create substitution structures with absolutely continuous spectrum and mixed spectral types, including constant-length substitutions.

## Key findings

- Structures exhibit purely absolutely continuous diffraction
- Examples include Fourier matrix-based substitutions
- Constant-length substitutions are achievable for any length

## Abstract

By generalising Rudin's construction of an aperiodic sequence, we derive new substitution-based structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.