Periodicity of the Spectrum of a Finite Union of Intervals

Abstract

A set , of Lebesgue measure 1, in the real line is called spectral if there is a set of real numbers such that the exponential functions e λ (x) = exp(2πiλx), λ ∈ , form a complete orthonormal system on L 2 ( ). Such a set is called a spectrum of . In this note we present a simplified proof of the fact that any spectrum of a set which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.

Size of orthogonal sets of exponentials for the disk

“…Writing H(a, ∆) for the hyperbola x 2 /a 2 − y 2 /b 2 = 1, with a 2 + b 2 = ∆ 2 , we consider the finite family of confocal hyperbolas (16) H k = H k 4 , ∆ , k = 0, 1, 2, . .…”

“…Research on spectral sets [22,13,20,19,3,2,16,10] has been driven for many years by a conjecture of Fuglede [6], sometimes called the Spectral Set Conjecture, which stated that a set Ω is spectral if and only if it is a translational tile. A set Ω is a translational tile if we can translate copies of Ω around and fill space without overlaps.…”

Untitled

“…The question of periodicity of one-dimensional spectra was explicitly raised in [Łaba 2002]. It was recently proved (first in [Bose and Madan 2011] and then a simplified proof was given in [Kolountzakis 2012]) that if is a finite union of intervals in the real line then any spectrum of is periodic. See also [Lagarias and Wang 1997], where periodicity of spectra and of tilings plays an important role.…”

“…Theorem 1.4 [Bose and Madan 2011;Kolountzakis 2012]. If D S n j D1 .a j ; b j / Â ‫ޒ‬ is a finite union of intervals of total length 1 and ƒ Â ‫ޒ‬ is a spectrum of , then there exists a positive integer T such that ƒ C T D ƒ.…”

“Spectral implies Tiling” for three intervals revisited

“…In particular, let us consider the generalized Vandermonde matrix which we get by taking those minors where the first three columns correspond to the left end-points of the set Ω and the 4th column is one of the right end point i.e., a minor obtained by choosing the 4th, 5th, and 6th columns of the matrix A and one of the first three columns. Thus we consider R (i 5 ,i 6…”