On the Rationality of the Spectrum

• Published in: The Journal of fourier analysis and applications Vol. 24; no. 4; pp. 1037 - 1047
• Main Authors: Bose, Debashish, Madan, Shobha
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2018; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Exponential functions; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Rationality; Signal,Image and Speech Processing; Tiling; Spectral sets; Rationality; Fuglede’s conjecture; 11D61; Secondary 42A99; Zeros of exponential polynomials; Primary 42C15; Spectrum; Recurrence sequences; 11B37
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-017-9552-8

Let Ω ⊂ R be a compact set with measure 1. If there exists a subset Λ ⊂ R such that the set of exponential functions E Λ : = { e λ ( x ) = e 2 π i λ x | Ω : λ ∈ Λ } is an orthonormal basis for L 2 ( Ω ) , then Λ is called a spectrum for the set Ω . A set Ω is said to tile R if there exists a set T such that Ω + T = R , the set T is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996 ) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011 ) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013 ). In this paper, we give some partial results to support the rationality of the spectrum.