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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2010, Volume 9, Issue 2, Pages 50–60 (Mi adm28)

RESEARCH ARTICLE

Perturbations of discrete lattices and almost periodic sets

Favorov Sergey, Kolbasina Yevgeniia

Mathematical School, Kharkov National University, Swobody sq.4, Kharkov, 61077, Ukraine

Abstract: A discrete set in the $p$-dimensional Euclidian space is almost periodic, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We propose to construct positive almost periodic discrete sets as an almost periodic perturbation of a full rank discrete lattice. Also we prove that each almost periodic discrete set on the real axes is an almost periodic perturbation of some arithmetic progression.
Next, we consider signed almost periodic discrete sets, i.e., when the signed measure with masses $+1$ or $-1$ at points of a discrete set is almost periodic. We construct a signed discrete set that is not almost periodic, while the corresponding signed measure is almost periodic in the sense of distributions. Also, we construct a signed almost periodic discrete set such that the measure with masses $+1$ at all points of the set is not almost periodic.

Keywords: perturbation of discrete lattice, almost periodic discrete set, signed discrete set, quasicrystals.

Full text: PDF file (218 kB)

Bibliographic databases:
MSC: 11K70, 52C07, 52C23
Revised: 10.11.2010
Language:

Citation: Favorov Sergey, Kolbasina Yevgeniia, “Perturbations of discrete lattices and almost periodic sets”, Algebra Discrete Math., 9:2 (2010), 50–60

Citation in format AMSBIB
\Bibitem{FavKol10} \by Favorov Sergey, Kolbasina Yevgeniia \paper Perturbations of discrete lattices and almost periodic sets \jour Algebra Discrete Math. \yr 2010 \vol 9 \issue 2 \pages 50--60 \mathnet{http://mi.mathnet.ru/adm28} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2808780} \zmath{https://zbmath.org/?q=an:1222.42011}