Quasicrystals with discrete support and spectrum

TL;DR

This paper demonstrates that measures with discrete support and spectrum need not have a periodic structure if the sets are merely discrete and closed, extending previous results that required uniform discreteness.

Contribution

It shows that the periodicity result for measures with discrete support and spectrum does not hold when the sets are only discrete and closed, broadening the understanding of quasicrystals.

Findings

Measures with discrete support and spectrum can lack periodicity if the sets are not uniformly discrete.

The previous periodicity result is limited to uniformly discrete sets, not all discrete closed sets.

The paper provides a counterexample to the periodicity in the more general setting.

Abstract

We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.