Abstract

Let D be a bounded domain in Rⁿ whose boundary has a Minkowski dimension α < n. Suppose that E_Λ={e^2πix•λ}_{λ in Λ}, Λ an infinite discrete subset of Rⁿ, is a frame of exponentials for L²(D), with frame constants A,B, A ≤ B. Then if

where C depends only on the ambient dimension n and |∂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.

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