How large are the spectral gaps?

Abstract

Let $D$ be a bounded domain in ${\mathbb{R}}^n$ whose boundary has a Minkowski dimension $\alpha <n$. Suppose that $E_{\Lambda }={\{e^{2\pi ix\cdot \lambda }\}}_{\lambda \in \Lambda }$, $\Lambda$ an infinite discrete subset of ${\mathbb{R}}^n$, is a frame of exponentials for $L^2(D)$, with frame constants $A,B$, $A\le B$. 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.