Diffraction of limit periodic point sets

TL;DR

This paper investigates the diffraction properties of limit periodic point sets, including examples like the period doubling sequence and chair tiling, revealing their pure point diffraction supported on a countably generated Fourier module.

Contribution

It provides explicit diffraction measure results for limit periodic sets with 2-adic internal spaces, expanding understanding of their aperiodic order and diffraction characteristics.

Findings

Diffraction measures for period doubling sequence derived.

Diffraction measures for chair tiling example derived.

Supports pure point diffraction on countably generated Fourier modules.

Abstract

Limit periodic point sets are aperiodic structures with pure point diffraction supported on a countably, but not finitely generated Fourier module that is based on a lattice and certain integer multiples of it. Examples are cut and project sets with p-adic internal spaces. We illustrate this by explicit results for the diffraction measures of two examples with 2-adic internal spaces. The first and well-known example is the period doubling sequence in one dimension, which is based on the period doubling substitution rule. The second example is a weighted planar point set that is derived from the classic chair tiling in the plane. It can be described as a fixed point of a block substitution rule.