Curved model sets and crystalline measures
type: Item
instance of: scholarly article
title: Curved model sets and crystalline measures
DOI: 10.1007/978-3-030-45847-8_17
author: Yves Meyer
publication date: 2024-09-16T00:00:00Z
review text: \textbf{Aim of the paper.} This paper constructs crystalline measures as curved versions of model sets. It relates diverse concepts in quasicrystals, crystallinn measures, and Fourier quasicrystals using a class of measures on tori pioneered by \textit{P. R. Ahern} [Mich. Math. J. 20, 33--37 (1973; Zbl 0265.32003)]. The author concludes Section 1 with the sentence ``Our approach does not yield new results but provides us with a new perspective on the achievements by Kurasov and Sarnak.''\N\N\textbf{Historical Background.} We give a brief historical background emphasizing the seminal contributions by Yves Meyer. Definitions are given in Section 3. In 1959 \textit{A. P. Guinand} [Acta Math. 101, 235--271 (1959; Zbl 0085.30102)] gave an ingenious construction of a crystalline measure on \(\mathbb R\). His derivation had an error that was corrected by \textit{Y. F. Meyer} [Proc. Natl. Acad. Sci. USA 113, No. 12, 3152--3158 (2016; Zbl 1367.28002); Rev. Mat. Iberoam. 33, No. 3, 1025--1036 (2017; Zbl 1384.42009)]. His measure is a Bohr almost periodic distributiion (see Lemma 2) but is not a Bohr almost measure because it is not translation bounded [\textit{Y. F. Meyer}, Proc. Natl. Acad. Sci. USA 113, No. 12, 3152--3158 (2016; Zbl 1367.28002); Bull. Hell. Math. Soc. 61, 11--20 (2017; Zbl 1425.42008)] (see page 394 lines 6--7). It is clearly a Fourier quasicrystal. We note that it is not a positve measure and that all positive Fourier quasicrystals are totally bounded.\N\NAperiodic planar tilings such as those of \textit{R. Penrose} [The role of aesthetics in pure and applied mathematical research'', Bull. Inst. Math. Appl. 10, 266--271 (1974)] and physical quasicrystals synthesized by \textit{D. Schechtman} et al. [Phys. Rev. Lett. 53, No. 1, 1951--1954 (1984; \url{doi:10.1103/PhysRevLett.53.1951})] are described by Meyer's model sets as explained by \textit{J. C. Lagarias} [Commun. Math. Phys. 179, No. 2, 365--376 (1996; Zbl 0858.52010)] and \textit{R. V. Moody} [NATO AS…
reviewed by: Wayne Lawton
Mathematics Subject Classification ID: 52C23
zbMATH DE Number: 7914256
zbMATH Keywords: Ahern measures
zbMATH Keywords: crystalline measures
zbMATH Keywords: quasicrystals
zbMATH Keywords: stable polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7914256