SCD Patterns Have Singular Diffraction

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.

11 pages, 2 figures; Accepted for Journal of Mathematical Physics

References in corpus (1)

• Diffraction of random tilings: some rigorous results

Cited by in corpus (6)

• Coincidence rotations of the root lattice $A_4$
• Hexagonal inflation tilings and planar monotiles
• On the notions of symmetry and aperiodicity for Delone sets
• About Substitution Tilings with Statistical Circular Symmetry
• Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules
• Almost sure recovery in quasi-periodic structures