2001/04/08 by Alex Iosevich, Steen Pedersen
Mathematics · #math.CA
published as How large are the spectral gaps?, (With S. Pedersen), Pacific J. Math., Volume 192, (2000), pp. 307-314
arxiv created 2001/04/08 · arxiv updated 2009/11/30
Let D be a bounded domain in \Bbb Rn whose boundary has a Minkowski dimension α<n. Suppose that EΛ= \e2 πi x ⋅ λ\λ∈ Λ, Λ an infinite discrete subset of \Bbb Rn, is a frame of exponentials for L2(D), with frame constants A,B, A ≤ B. Then if R ≥ C(\fracB|∂ D|αA|D| )^ (1)/(n-α), 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.