Bohr and Besicovitch almost periodic discrete sets and quasicrystals

2010/11/17 by Favorov, Sergey
Materials Science · #42A75 #52C07 #52C23 #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1011.4036

openalex publication_date 2010/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A discrete set A in the 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 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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