Bohr and Besicovitch almost periodic discrete sets and quasicrystals

TL;DR

This paper explores the properties of almost periodic discrete sets in Euclidean space, particularly when their difference set is discrete, revealing structural insights related to lattices and quasicrystals.

Contribution

It characterizes Bohr and Besicovitch almost periodic sets with discrete differences, showing their relation to lattices and their structural composition.

Findings

Bohr almost periodic sets are unions of finitely many lattice translates.

Besicovitch almost periodic sets intersect lattice translates significantly.

Structural properties connect almost periodicity with quasicrystal arrangements.

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.