SCD Patterns Have Singular Diffraction

TL;DR

This paper analyzes the diffraction spectra of SCD tilings, revealing their pure point and singular continuous components, and providing detailed spectral structure insights for specific cases.

Contribution

It determines the diffraction spectra of SCD tilings, showing their singular diffraction patterns and spectral components, which were previously not well understood.

Findings

No absolutely continuous diffraction component.

Pure point diffraction on the z-axis.

Support on concentric cylinder surfaces.

Abstract

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that they have no absolutely continuous part, that they have a uniformly discrete pure point part on the z-axis, and that they are otherwise supported on a set of concentric cylinder surfaces around this axis. For SCD tilings with additional properties, more detailed results are given.