Pure Point Dynamical and Diffraction Spectra

TL;DR

This paper proves the equivalence between pure point diffraction and pure point dynamical spectrum for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies, linking spectral properties to geometric structure.

Contribution

It establishes a fundamental equivalence between diffraction and dynamical spectra for a broad class of aperiodic point sets, advancing understanding in mathematical diffraction theory.

Findings

Pure point diffraction implies pure point dynamical spectrum.

Pure point dynamical spectrum implies pure point diffraction.

The equivalence holds for multi-colored Delone sets with finite local complexity and uniform cluster frequencies.

Abstract

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.