A short guide to pure point diffraction in cut-and-project sets

Abstract

We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochner's theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples.

25 pages, 3 figures, v2: presentation improved

References in corpus (7)

• Soft quasicrystals - Why are they stable?
• Dynamics on the graph of the torus parametrisation
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Ergodic properties of visible lattice points
• Almost Periodic Measures and Meyer Sets
• Toeplitz flows and model sets
• $\mathcal{B}$-free sets and dynamics

Cited by in corpus (16)

• On the Fourier Analysis of Measures with Meyer Set Support
• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• Leptin densities in amenable groups
• Modulated crystals and almost periodic measures
• On sampling and interpolation by model sets
• A note on measures vanishing at infinity
• Model sets with precompact Borel windows
• Inflation versus projection sets in aperiodic systems: The role of the window in averaging and diffraction
• On the Fourier Transformability of Strongly Almost Periodic Measures
• On the Garden of Eden theorem for B-free subshifts
• Why do Meyer sets diffract?
• Dynamical spectrum of power-free integers in quadratic number fields and beyond
• Measures on the Spectra of Algebraic Integers
• Diffraction of return time measures
• Semi-measures and their Fourier transform
• Fourier Transformable Measures with Meyer set support and their lift to the cut and project scheme