# Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

## TL;DR

This paper analyzes a simple non-Pisot substitution tiling, demonstrating through renormalization that its diffraction pattern is purely singular continuous, with no absolutely continuous component, apart from a trivial Bragg peak.

## Contribution

It provides a detailed geometric and algebraic analysis of a non-Pisot tiling, establishing the nature of its diffraction measure as purely singular continuous.

## Key findings

- Diffraction measure has no absolutely continuous component.
- Diffraction is purely singular continuous except for a trivial Bragg peak.
- Derived geometric and algebraic properties of the underlying Delone dynamical system.

## Abstract

One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well.