Bohr and Besicovitch almost periodic discrete sets and quasicrystals

Details Bohr and Besicovitch almost periodic discrete sets and quasicrystals Bohr and Besicovitch almost periodic discrete sets and quasicrystals

2010-11-17

arxiv.org/abs/1011.4036v1

Mathematics

Metric Geometry

5 pages, bibliography 12 items

Scientific paper

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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