Measures with uniformly discrete support and spectrum (Q384089)

A set \(\Lambda\subset\mathbb{R}^n\) is called uniformly discrete if NEWLINE\[NEWLINE \inf\{|\lambda-\lambda'|: \lambda,\lambda'\in \Lambda, \lambda\neq \lambda'\} > 0. NEWLINE\]NEWLINENEWLINENEWLINENEWLINEA measure \(\mu\) on \(\mathbb{R}^n\) is said to be supported on a set \(\Lambda \subset \mathbb{R}^n\) if NEWLINE\[NEWLINE \mu=\sum_{\lambda\in\Lambda} \mu(\lambda)\delta_\lambda, NEWLINE\]NEWLINE NEWLINEwhere \(\mu(\lambda)\neq 0\), \(\lambda\in \Lambda,\) and \(\delta_\lambda\) denotes the Dirac measure.NEWLINENEWLINENEWLINEIf a Fourier transform of a measure \(\mu\) on \(\mathbb{R}^n\) is also a measure and \(\widehat{\mu}\) is supported on a set \(S\subset\mathbb{R}^n\), then \(S\) is called the spectrum of \(\mu\).NEWLINENEWLINENEWLINEIn the paper the authors give a sketch of a proof of the following result.NEWLINENEWLINENEWLINE{Theorem 1.} If a measure \(\mu\) on \(\mathbb{R}\) has both uniformly discrete support and spectrum, then the support of \(\mu\) is contained in a finite union of translates of a certain lattice.NEWLINENEWLINENEWLINETheorem 1 confirms the conjecture of \textit{J. C. Lagarias} [CRM Monogr. Ser. 13, 61--93 (2000; Zbl 1161.52312)].NEWLINENEWLINEFor the multidimensional case the following result is given.NEWLINENEWLINENEWLINE{Theorem 2.} If a measure \(\mu\) on \(\mathbb{R}^n\), \(n>1\), has both uniformly discrete support and spectrum and if additionally it is known that \(S-S\) is uniformly discrete (here \(S\) denotes the spectrum of \(\mu\)), then the support of \(\mu\) is contained in a finite union of translates of a certain lattice.