Any discrete almost periodic set of finite type is an ideal crystal

TL;DR

The paper proves that any discrete almost periodic set with a discrete difference set in Euclidean space must be composed of finitely many translates of a lattice, linking almost periodicity with crystalline structures.

Contribution

It establishes a new characterization of discrete almost periodic sets of finite type as finite unions of lattice translates, bridging almost periodicity and crystal structures.

Findings

Discrete almost periodic sets with discrete difference sets are finite unions of lattice translates.

The result characterizes the structure of such sets as ideal crystals.

Provides a link between almost periodicity and crystallography in Euclidean spaces.

Abstract

A discrete set 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 prove the following result: if A is a discrete almost periodic set and the set A-A is discrete, then A consists of a finite number of translates of a full rank lattice.