Existence of quasicrystals and universal stable sampling and interpolation in LCA groups

TL;DR

This paper characterizes the presence of quasicrystals in locally compact abelian groups and extends the concepts of stable sampling and interpolation from Euclidean spaces to these groups, revealing their universality.

Contribution

It provides a complete characterization of quasicrystals in LCA groups and extends stable sampling and interpolation results to these groups.

Findings

Characterization of all LCA groups containing quasicrystals.

Construction method for quasicrystals via lattice and cut-and-project scheme.

Extension of Meyer and Matei's results to LCA groups, showing universality of simple quasicrystals.

Abstract

We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group G admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the n-dimensional Euclidean space can be extended to G. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in generalized Paley-Wiener spaces.