Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Bohr and Besicovitch almost periodic discrete sets and quasicrystals

Author: S. Favorov
Journal: Proc. Amer. Math. Soc. 140 (2012), 1761-1767
MSC (2010): Primary 52C23; Secondary 42A75, 52C07
Posted: August 22, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A discrete set in the Euclidean space is almost periodic if the measure with the unit masses at points of the set is almost periodic in the weak sense. We investigate properties of such sets in the case when is discrete. In particular, if is a Bohr almost periodic set, we prove that is a union of a finite number of translates of a certain full-rank lattice. If is a Besicovitch almost periodic set, then there exists a full-rank lattice such that in most cases a nonempty intersection of its translate with is large.

References

Bibliography

1. J. Andres, A. M. Bersani, and R. F. Grande, Hierarchy of almost-periodic function spaces, Rend. Mat. Appl. (7) 26 (2006), no. 2, 121–188. MR 2275292 (2008b:43010)
2. Michael Baake, Daniel Lenz, and Robert V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 341–382. MR 2308136 (2008f:37007), http://dx.doi.org/10.1017/S0143385706000800
3. A. S. Besicovitch, Almost periodic functions, Dover Publications Inc., New York, 1955. MR 0068029 (16,817a)
4. S. Ju. Favorov, Sunyer-i-Balaguer’s almost elliptic functions and Yosida’s normal functions, J. Anal. Math. 104 (2008), 307–340. MR 2403439 (2009f:30066), http://dx.doi.org/10.1007/s11854-008-0026-4
5. S. Favorov and Ye. Kolbasina, Almost periodic discrete sets, Zh. Mat. Fiz. Anal. Geom. 6 (2010), no. 1, 34–47, 135 (English, with English and Russian summaries). MR 2655763 (2011g:42011)
6. Sergey Favorov and Yevgeniia Kolbasina, Perturbations of discrete lattices and almost periodic sets, Algebra Discrete Math. 9 (2010), no. 2, 50–60. MR 2808780
7. Sergey Favorov, Alexander Rashkovskii, and Lev Ronkin, Almost periodic divisors in a strip, J. Anal. Math. 74 (1998), 325–345. MR 1631678 (99k:42016), http://dx.doi.org/10.1007/BF02819455
8. N. D. Parfyonova and S. Yu. Favorov, Meromorphic almost periodic functions, Mat. Stud. 13 (2000), no. 2, 190–198 (English, with English and Russian summaries). Dedicated to A. A. Gol′dberg on the occasion of his 70th anniversary. MR 1776544 (2001e:30045)
9. Jean-Baptiste Gouéré, Quasicrystals and almost periodicity, Comm. Math. Phys. 255 (2005), no. 3, 655–681. MR 2135448 (2006j:52026), http://dx.doi.org/10.1007/s00220-004-1271-8
10. J. C. Lagarias, Meyer’s concept of quasicrystal and quasiregular sets, Comm. Math. Phys. 179 (1996), no. 2, 365–376. MR 1400744 (97g:52049)
11. J. C. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom. 21 (1999), no. 2, 161–191. MR 1668082 (99m:52029), http://dx.doi.org/10.1007/PL00009413
12. Jeffrey C. Lagarias, Mathematical quasicrystals and the problem of diffraction, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 61–93. MR 1798989 (2001m:52032)
13. B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888 (81k:30011)
14. Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam, 1972. North-Holland Mathematical Library, Vol. 2. MR 0485769 (58 #5579)
15. Robert V. Moody, Meyer sets and their duals, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, Kluwer Acad. Publ., Dordrecht, 1997, pp. 403–441. MR 1460032 (98e:52029)
16. Robert V. Moody and Nicolae Strungaru, Point sets and dynamical systems in the autocorrelation topology, Canad. Math. Bull. 47 (2004), no. 1, 82–99. MR 2032271 (2005e:52031), http://dx.doi.org/10.4153/CMB-2004-010-8
17. E. Arthur Robinson Jr., A Halmos-von Neumann theorem for model sets, and almost automorphic dynamical systems, Dynamics, ergodic theory, and geometry, Math. Sci. Res. Inst. Publ., vol. 54, Cambridge Univ. Press, Cambridge, 2007, pp. 243–272. MR 2369449 (2010a:37019), http://dx.doi.org/10.1017/CBO9780511755187.010
18. L. I. Ronkin, Almost periodic generalized functions in tube domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 247 (1997), no. Issled. po Linein. Oper. i Teor. Funkts. 25, 210–236, 303 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 101 (2000), no. 3, 3172–3189. MR 1692679 (2000m:46090), http://dx.doi.org/10.1007/BF02673742
19. Martin Schlottmann, Generalized model sets and dynamical systems, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 143–159. MR 1798991 (2001k:52035)
20. H. Tornehave, Systems of zeros of holomorphic almost periodic functions, Kobenhavns Universitet Matematisk Institut, preprint No. 30, 1988, 52 pp.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|