Modulated crystals and almost periodic measures
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Modulated crystals and almost periodic measures Jeong-Yup Lee1,2 · Daniel Lenz3 · Christoph Richard4 Nicolae Strungaru6,7
· Bernd Sing5 ·
Received: 13 November 2019 / Revised: 23 October 2020 / Accepted: 30 October 2020 / Published online: 17 November 2020 © The Author(s) 2020
Abstract Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations. Keywords Diffraction · Modulated crystal · Model set · Cut-and-project scheme · Almost periodic measure Mathematics Subject Classification 52C23 · 37A25 · 37B10 · 37B50
1 Introduction In 1964, Brouns et al. performed an X-ray diffraction experiment on a washing soda crystal, which was expected to behave like a perfect crystal. Surprisingly, a slight anomaly in its diffraction pattern was found [17]. It was later suggested to describe such anomalies by modulated crystals [21,31]. Over the years, the crystallography of such structures has been worked out and has found many applications including protein crystallography [43]. Numerous important contributions to the field have been made by Ted Janssen [64]. Recent overviews of modulated structures and of other types of aperiodic crystals are Janssen [34] and Janner and Janssen [32]. See also the monographs by Janssen et al. [33] and by van Smaalen [67] for a detailed discussion of aperiodic crystals from physical and crystallographic perspectives. On the mathematical side, Bombieri and Taylor [15] suggested studying modulated lattices in 1985, as they appeared to have properties similar to quasicrystals, a funda-
We dedicate this work to Michael Baake on the occasion of his 60th birthday. Extended author information available on the last page of the article
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mentally different type of aperiodic crystal that had been discovered shortly before. Models for quasicrystals were obtained by the so-called cut-and-project construction [23,40]. De Bruijn introduced modulated quasicrystals in Euclidean space [20], a class of point sets that is nowadays subsumed by so-called deformed weighted model sets. Whereas this class comprises weighted cut-and-project sets, de Bruijn showed that it also includes certain modulated lattices [20, Sec. 5,6]. De Bruijn’s Fourier analysis relied on a particular class of smooth weight functions of unbounded support, which cannot readily be extended beyond Euclidean space. Deformed weighted model sets have further been studied by Hof [30] and by Bernuau and Duneau [13], who coined the name, see also [32]. In the meantime, it was realized that cut-and-projects sets are in fact model sets as introduced earl
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