Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

TL;DR

This paper proves that Delone sets with uniform cluster frequencies and pure point diffraction have zero entropy, linking diffraction properties to complexity measures in aperiodic order.

Contribution

It establishes that pure point diffraction combined with uniform cluster frequencies implies zero topological and patch counting entropy in Delone sets.

Findings

Pure point diffractive Delone sets with uniform cluster frequencies have zero entropy.

Patch counting entropy is zero under certain growth conditions of the repetitivity function.

The results connect diffraction properties with complexity measures in aperiodic structures.

Abstract

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.