How large are the spectral gaps?

Abstract

Let $D$ be a bounded domain in ${\Bbb R}^n$ whose boundary has a Minkowski dimension $\alpha<n$. Suppose that $E_{\Lambda}= {\{e^{2 \pi i x \cdot \lambda}\}}_{\lambda \in \Lambda}$, $\Lambda$ an infinite discrete subset of ${\Bbb R}^n$, is a frame of exponentials for $L^2(D)$, with frame constants $A,B$, $A \leq B$. Then if $$ R \ge C{(\frac{{B|\partial D|}_{\alpha}}{A|D|} )}^ {\frac{1}{n-\alpha}},$$ where $C$ depends only on the ambient dimension $n$ and ${|\partial D|}_{\alpha}$ denotes the Minkowski content, then every cube of sidelength $R$ contains at least one element of $\Lambda$. We give examples that illustrate the extent to which our estimates are sharp.