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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
Published electronically: August 22, 2011
MathSciNet review: 2869161
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Abstract: A discrete set $ A$ 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 $ A-A$ is discrete. In particular, if $ A$ is a Bohr almost periodic set, we prove that $ A$ is a union of a finite number of translates of a certain full-rank lattice. If $ A$ 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 $ A$ is large.


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Additional Information

S. Favorov
Affiliation: Mathematical School, Kharkov National University, Swobody sq. 4, Kharkov, 61077 Ukraine
Email: sfavorov@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11046-3
Keywords: Quasicrystals, Bohr almost periodic set, Besicovitch almost periodic set, ideal crystal
Received by editor(s): November 11, 2010
Received by editor(s) in revised form: January 11, 2011
Published electronically: August 22, 2011
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.