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Why do (weak) Meyer sets diffract?

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Abstract

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.

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Acknowledgements

The work was supported by NSERC with grant 2020-00038, and we are greatly thankful for all the support. The generalization to FCDM functions, and its application were suggested by Daniel Lenz, and we are grateful for his insightful feedback. We would like to thank Anna Klick for creating Fig. 1 in Maple, and for some suggestions which improved the quality of the manuscript. We would like to thank Michael Baake for many helpful suggestions, and Adam Humeniuk for carefully reading the manuscript and for many suggestions which helped improve the paper. Finally, we would like to thank the two anonymous referees for numerous suggestions which improved the quality of this manuscript.

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  1. Department of Mathematical Sciences, MacEwan University, 10700 – 104 Avenue, Edmonton, AB, T5J 4S2, Canada

    Nicolae Strungaru

  2. Institute of Mathematics “Simon Stoilow”, Bucharest, Romania

    Nicolae Strungaru

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  1. Nicolae Strungaru
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Correspondence to Nicolae Strungaru.

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We dedicate this work to Daniel Lenz on the occasion of his 50th birthday.

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Strungaru, N. Why do (weak) Meyer sets diffract?. Lett Math Phys 113, 52 (2023). https://doi.org/10.1007/s11005-023-01676-w

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