How large are the spectral gaps?

Alex Iosevich and Steen Pedersen

Vol. 192 (2000), No. 2, 307–314

DOI: 10.2140/pjm.2000.192.307

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∈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.

Milestones

Received: 25 June 1998

Published: 1 February 2000