Why do (weak) Meyer sets diffract? (English) Zbl 1526.43003
Summary: Given a weak model set in a locally compact abelian group, we construct a relatively dense set of common Bragg peaks for all its subsets that have non-trivial Bragg spectrum. Next, we construct a relatively dense set of common norm almost periods for the diffraction, pure point, absolutely continuous and singular continuous spectrum, respectively, of all its subsets. We use the Fibonacci model set to illustrate these phenomena. We extend all these results to arbitrary translation bounded weighted Dirac combs supported within some Meyer set. We complete the paper by discussing extensions of the existence of the generalized Eberlein decomposition for measures supported within some Meyer set.
MSC:
| 43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |
Keywords:
Meyer sets; diffraction; Bragg spectrum; Fibonacci model set; continuous diffraction spectraReferences:
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