Harmonic analysis on tori (Q1093835)

The authors consider a measurable subset \(\Omega \subseteq {\mathbb{R}}^ n\) with spectral set \(\Lambda\). It is not assumed that \(\Lambda\) is a lattice. Instead it is assumed that \(\Lambda\) is the union of shifts of the dual of a lattice. Under these circumstances, necessary and sufficient conditions are derived for \(\Omega\) to be the fundamental domain of some lattice in \({\mathbb{R}}^ n\). A variety of applications are supplied.