How large are the spectral gaps?

Abstract

Let $D$ be a bounded domain in ${\Bbb R}^n$ whose boundary has a Minkowski dimension $α<n$. Suppose that $E_Λ= {\{e^{2 πi x \cdot λ}\}}_{λ\in Λ}$, $Λ$ 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 $$ R \ge C{(\frac{{B|\partial D|}_α}{A|D|} )}^ {\frac{1}{n-α}},$$ where $C$ depends only on the ambient dimension $n$ and ${|\partial D|}_α$ denotes the Minkowski content, then every cube of sidelength $R$ contains at least one element of $Λ$. We give examples that illustrate the extent to which our estimates are sharp.