光谱间隙有多大？
How large are the spectral gaps?
摘要：
设$D$是${\Bbb R}^n$中的一个有界域，其边界具有Minkowski维数$α<n$ .假设$E_Λ= {\{e^{2 πi x \cdot λ}\}}_{λ\in Λ}$，$Λ$是${\Bbb R}^n$的无限离散子集，是$L^2(D)$的指数标架，标架常数为$A,B$，$A \leq B$ .然后，如果$$ R \ge C{(\frac{{B|\partial D|}_α}{A|D|} )}^ {\frac{1}{n-α}},$$，其中$C$仅取决于环境维度$n$，并且${|\partial D|}_α$表示Minkowski内容，则侧长$R$的每个立方体包含$Λ$的至少一个元素。我们给出的例子说明了我们的估计的准确程度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
