Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-16T15:32:48.515Z Has data issue: false hasContentIssue false

Chapter 5 - Almost Periodic Pure Point Measures

Published online by Cambridge University Press:  26 October 2017

By
Nicolae Strungaru
Edited by
Michael Baake  and
Uwe Grimm
Nicolae Strungaru
Affiliation:
Dept. of Mathematics and Statistics MacEwan University Edmonton, Canada
Michael Baake
Affiliation:
Universität Bielefeld, Germany
Uwe Grimm
Affiliation:
The Open University, Milton Keynes
Get access

Summary

Motivated by the general structure of mathematical quasicrystals, we construct a cut and project scheme from a family of sets that satisfy some general and fairly natural properties. We use this construction to characterise weighted Dirac combs derived from such a scheme via continuous functions on the internal group in terms of almost periodicity. In particular, we discuss weighted Dirac combs where the internal function is compactly supported. More generally, using the same cut and project construction for ε-dual sets, we characterise Meyer sets in locally compact Abelian groups.

Introduction

In 1984, D. Shechtman et al. announced the discovery (from 1982) of a solid with an unusual diffraction pattern [42]. While the diffraction was similar to that of a periodic crystal, exhibiting only bright spots (called Bragg peaks) and little or no diffuse background, it also showed fivefold symmetry— a phenomenom which is impossible in a perfect (periodic) crystal in 3-space; see [AO1, Cor. 3.1]. Solids with such unusual (non-crystallographic) symmetries are called (genuine) quasicrystals. Such a quasicrystal cannot repeat periodically in a set of directions that span the ambient space. Yet, in order to produce many Bragg peaks, a large number of its local motives (or patches) need to repeat in a highly ordered and coherent fashion.

Twelve years earlier, Y. Meyer [29] introduced the concept of harmonious sets. These sets exhibit long-range order and are usually non-periodic. Meyer also introduced cut and project schemes as a simple way of generating such sets and studied the relationship between harmonious sets and model sets; see below for more on these notions and their connections with the projection method. In the context of icosahedral structures, the latter was independently discovered by P. Kramer and one of his students [19, 21]; see the Epilogue to this volume [20] for a more detailed account and [AO1, Ch. 7] for general background. The relevance of Meyer's work to quasicrystals and its relations to the other formulations was realised only in the 1990s; see [30, 22].

Type
Chapter
Information
Aperiodic Order , pp. 271 - 342
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AO1] Baake, M. and Grimm, U. (2013) Aperiodic Order. Vol. 1: A Mathematical Invitation (Cambridge University Press, Cambridge).
[1] Argabright, L.N. and Gil de Lamadrid, J. (1974) Fourier analysis of unbounded measures on locally compact Abelian groups, Memoirs Amer. Math. Soc., no. 145 (AMS, Providence, RI).
[2] Aujogue, J.-B. (2016) Pure point/continuous decomposition of translation-bounded measures and diffraction, Preprint arXiv:1510.06381.
[3] Baake, M. and Huck, C. (2015) Ergodic properties of visible lattice points, Proc. Steklov Inst. Math. 288, 165–188. arXiv:1501.01198.Google Scholar
[4] Baake, M., Huck, C. and Strungaru, N. (2017) On weak model sets of extremal density, Indag. Math. 28, 3–31. arXiv:1512.07129.Google Scholar
[5] Baake, M. and Lenz, D. (2004) Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Th. & Dynam. Syst. 24, 1867–1893. arXiv:math.DS/0302231.CrossRefGoogle Scholar
[6] Baake, M., Lenz, D. and Moody, R.V. (2007) Characterisation of model sets by dynamical systems, Ergodic Th. & Dynam. Syst. 27, 341–382. arXiv:math.DS/0511648.Google Scholar
[7] Baake, M. and Moody, R.V. (2004) Weighted Dirac combs with pure point diffraction, J. reine angew. Math. (Crelle) 573, 61–94. arXiv:math.MG/0203030.CrossRefGoogle Scholar
[8] Baake, M., Moody, R.V. and Pleasants, P.A.B. (2000) Diffraction from visible lattice points and k-th power free integers, Discr. Math. 221, 3–42. arXiv:math.MG/9906132.Google Scholar
[9] Berg, C. and Forst, G. (1975) Potential Theory on Locally Compact Abelian Groups (Springer, Berlin).
[10] Bourbaki, N. (1966) General Topology, parts I and II (Hermann, Paris).
[11] Eberlein, W.F. (1949) Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67, 217–224.CrossRefGoogle Scholar
[12] Gil de Lamadrid, J. and Argabright, L.N. (1990) Almost periodic measures, Memoirs Amer. Math. Soc. 85, no. 428 (AMS, Providence, RI).
[13] Hewitt, E. and Ross, K.A. (1997) Abstract Harmonic Analysis I, 2nd ed., corr. 3rd printing (Springer, New York).
[14] Hof, A. (1995) On diffraction by aperiodic structures, Commun. Math. Phys. 169, 25–43.CrossRefGoogle Scholar
[15] Huck, C. and Baake, M. (2014) Dynamical properties of k-free lattice points, Acta Phys. Pol. A 126, 482–485. arXiv:1402.2202.Google Scholar
[16] Huck, C. and Richard, C. (2015) On pattern entropy of weak model sets, Discr. Comput. Geom. 54, 741–757. arXiv:1412.6307.Google Scholar
[17] Kellendonk, J. and Sadun, L. (2014) Meyer sets,topological eigenvalues,and Cantor fiber bundles, J. London Math. Soc. 89, 114–130. arXiv:1211.2250.CrossRefGoogle Scholar
[18] Keller, G. and Richard, C. (2017) Dynamics on the graph of the torus parametrization, Ergodic Th. & Dynam. Syst., in press. arXiv:1511.06137.
[19] Kramer, P. (1982) Non-periodic central space filling with icosahedral symmetry using copies of seven elementary cells, Acta Cryst. A 38, 257–264.Google Scholar
[20] Kramer, P. (2017) Gateways towards quasicrystals. In Aperiodic Order. Vol. 2: Crystallography and Almost Periodicity (this volume), Baake, M. and Grimm, U. eds., pp. 363–380 (Cambridge University Press, Cambridge).
[21] Kramer, P. and Neri, R. (1984) On periodic and non-periodic space fillings of Em obtained by projection,Acta Cryst. A 40, 580–587 and Acta Cryst. A 41(1985), 619 (erratum).Google Scholar
[22] Lagarias, J.C. (1996) Meyer's concept of quasicrystal and quasiregular sets, Commun. Math. Phys. 179, 365–376.CrossRefGoogle Scholar
[23] Lagarias, J.C. (2000) Mathematical quasicrystals and the problem of diffraction. In Directions in Mathematical Quasicrystals, CRM Monograph Series, vol. 13, Baake, M. and Moody|R.V eds., pp. 61–93 (AMS, Providence, RI).
[24] Lenz, D. (2009) Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks, Commun. Math. Phys. 87, 225–258. arXiv:math.ph/0608026.Google Scholar
[25] Lenz, D. and Richard, C. (2007) Pure point diffraction and cut and project schemes for measures: The smooth case, Math. Z. 256, 347–378. arXiv:math.DS/0603453.CrossRefGoogle Scholar
[26] Lenz, D. and Strungaru, N. (2009) Pure point spectrum for measurable dynamical systems on locally compact Abelian groups, J. Math. Pures Appl. 92, 323–341. arXiv:0704.2498.Google Scholar
[27] Lenz, D. and Strungaru, N. (2016) Note on the set of Bragg peaks with high intensity, Ann. H. Poincarè 17, 673–687. arXiv:1412.7377.Google Scholar
[28] Lenz, D. and Strungaru, N. (2016) On weakly almost periodic measures, Preprint arXiv:1609.08219.
[29] Meyer, Y. (1972) Algebraic Numbers and Harmonic Analysis (North Holland, Amsterdam).
[30] Moody, R.V. (1997) Meyer sets and their duals. In The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489, Moody, R.V. eds., pp. 403–441 (Kluwer, Dordrecht).
[31] Moody, R.V. (2002) Uniform distribution in model sets, Can. Math. Bull. 45, 123–130.CrossRefGoogle Scholar
[32] Moody, R.V. and Strungaru, N. (2004) Discrete sets and dynamical systems in the autocorrelation topology, Can. Math. Bull. 47, 82–99.CrossRefGoogle Scholar
[33] Moody, R.V. and Strungaru, N. (2017) Almost periodic measures and their Fourier transforms. In Aperiodic Order. Vol. 2: Crystallography and Almost Periodicity (this volume), Baake, M. and Grimm, U. eds., pp. 173–270 (Cambridge University Press, Cambridge).
[34] Pleasants, P.A.B. and Huck, C. (2013) Entropy and diffraction of the k-free points in n-dimensional lattices, Discr. Comput. Geom. 50, 39–68. arXiv:1112.1629.Google Scholar
[35] Reed, M. and Simon, B. (1980) Methods of Modern Mathematical Physics I: Functional Analysis, 2nd ed. (Academic Press, San Diego, CA).
[36] Reiter, H. and Stegeman, J.D. (2001) Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press, Oxford).
[37] Richard, C. (2003) Dense Dirac combs in Euclidean space with pure point diffraction, J. Math. Phys. 44, 4436–4449. arXiv:math-ph/0302049.CrossRefGoogle Scholar
[38] Richard, C. and Strungaru, N. (2015) Pure point diffraction and Poisson summation, Preprint arXiv:1512.00912.
[39] Rudin, W. (1962) Fourier Analysis on Groups (Wiley, New York).
[40] Schlottmann, M. (1998) Cut-and-project sets in locally compact Abelian groups. In Quasicrystals and Discrete Geometry, Fields Institute Monographs, vol. 10, Patera, J. eds., pp. 247–264 (AMS, Providence, RI).
[41] Schreiber, J.-P. (1973) Approximations diophantiennes et problèmes additifs dans les groupes abèliens localement compacts, Bull. Soc. Math. France 101, 297–332.CrossRefGoogle Scholar
[42] Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W. (1984) Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53, 1951–1953.CrossRefGoogle Scholar
[43] Strungaru, N. (2005) Almost periodic measures and long-range order in Meyer sets, Discr. Comput. Geom. 33, 483–505.CrossRefGoogle Scholar
[44] Strungaru, N. (2013) On the Bragg diffraction spectra of a Meyer set, Can. J. Math. 65, 675–701. arXiv:1003.3019.Google Scholar
[45] Strungaru, N. (2013) Almost periodic measures and Bragg diffraction, J. Phys. A: Math. Theor. 46, 125205: 1–11. arXiv:1209.2168.CrossRefGoogle Scholar
[46] Strungaru, N. (2014) On weighted Dirac combs supported inside model sets, J. Phys. A: Math. Theor. 47, 335202: 1–19. arXiv:1309.7947.CrossRefGoogle Scholar
[47] Strungaru, N. and Terauds, V. (2016) Diffraction theory and almost periodic distributions, J. Stat. Phys. 164, 1183–1216. arXiv:1603.04796.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×