Dense Dirac combs in Euclidean space with pure point diffraction. (English) Zbl 1062.52022
Summary: Regular model sets, describing the point positions of ideal quasicrystallographic tilings, are mathematical models of quasicrystals. An important result in mathematical diffraction theory of regular model sets, which are defined on locally compact abelian groups, is the pure pointedness of the diffraction spectrum. We derive an extension of this result, valid for dense point sets in Euclidean space, which is motivated by the study of quasicrystallographic random tilings.
MSC:
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 11K41 | Continuous, \(p\)-adic and abstract analogues |
References:
| [1] | Baake, J. Stat. Phys. 99 pp 219– (2000) |
| [2] | Höffe |
| [3] | Baake, J. Phys. A 31 pp 9023– (1998) |
| [4] | Moody |
| [5] | Baake, M. and Moody, R.V., ”Self-similar measures for quasicrystals,” inDirections in Mathematical Quasicrystals, edited by M. Baake and R.V. Moody, CRM Monograph Series, Vol. 13 (AMS, Providence, RI, 2000), pp. 1–42; · Zbl 0972.52013 |
| [6] | Moody |
| [7] | Baake (2002) |
| [8] | Baake, Discrete Math. 221 pp 3– (2000) |
| [9] | Pleasants |
| [10] | Baake |
| [11] | Bernuau, G. and Duneau, M., ”Fourier analysis of deformed model sets,” inDirections in Mathematical Quasicrystals, edited by M. Baake and R.V. Moody, CRM Monograph Series, Vol. 13 (AMS, Providence, RI, 2000), pp. 43–60. · Zbl 1001.42007 |
| [12] | Bombieri, J. Phys. Colloq. 47 pp 19– (1986) |
| [13] | Córdoba, Lett. Math. Phys. 17 pp 191– (1989) |
| [14] | Dworkin, J. Math. Phys. 34 pp 2965– (1993) |
| [15] | Gil de Lamadrid, J. and Argabright, L.N., ”Almost periodic measures,” Memoirs of the AMS, Vol. 428 (AMS, Providence, RI, 1990). · Zbl 0719.43006 |
| [16] | Henley, C.L., ”Random tiling models,” inQuasicrystals: The State of the Art, 2nd ed. edited by D.P. DiVincenzo and P.J. Steinhardt, Series on Directions in Condensed Matter Physics, Vol. 16 (World Scientific, Singapore, 1999), pp. 459–560. |
| [17] | Hof, Commun. Math. Phys. 169 pp 25– (1995) |
| [18] | Hof, A., ”Uniform distribution and the projection method,” inQuasicrystals and Discrete Geometry, edited by J. Patera, Fields Institute Monographs, Vol. 10 (AMS, Providence, RI, 1998), pp. 201–206. · Zbl 0913.28012 |
| [19] | Höffe, M., ”Diffraktionstheorie stochastischer Parkettierungen,” Dissertation, Universität Tübingen, Shaker, Aachen, 2001. |
| [20] | Höffe, Z. Kristallogr. 215 pp 441– (2000) |
| [21] | Baake |
| [22] | Kuipers, L. and Niederreiter, H.,Uniform Distribution of Sequences(Wiley, New York, 1974). · Zbl 0281.10001 |
| [23] | Külske (2001) |
| [24] | Lagarias, J.C., ”Mathematical quasicrystals and the problem of diffraction,” inDirections in Mathematical Quasicrystals, edited by M. Baake and R.V. Moody, CRM Monograph Series, Vol. 13 (AMS, Providence, RI, 2000), pp. 61–93. · Zbl 1161.52312 |
| [25] | Lenz, D. (private communication, 2002). |
| [26] | Moody, R.V., ”Meyer sets and their duals,” inThe Mathematics of Long-Range Aperiodic Order, edited by R.V. Moody, NATO ASI Series C 489 (Kluwer, Dordrecht, 1997), pp. 403–441. · Zbl 0880.43008 |
| [27] | Reed, M. and Simon, B.,Methods of Modern Mathematical Physics. I: Functional Analysis, 2nd ed. (Academic, San Diego, CA, 1980). · Zbl 0459.46001 |
| [28] | Richard, J. Phys. A 32 pp 8823– (1999) |
| [29] | Richard |
| [30] | Richard, J. Phys. A 31 pp 6385– (1998) |
| [31] | Baake |
| [32] | Schlottmann, M., ”Cut-and-project sets in locally compact Abelian groups,” inQuasicrystals and Discrete Geometry, edited by J. Patera, Fields Institute Monographs, Vol. 10 (AMS, Providence, RI, 1998), pp. 247–264. · Zbl 0912.22002 |
| [33] | Schlottmann, M., ”Generalized model sets and dynamical systems,” inDirections in Mathematical Quasicrystals, edited by M. Baake and R.V. Moody, CRM Monograph Series, Vol. 13 (AMS, Providence, RI, 2000), pp. 143–159. · Zbl 0984.37005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
