Spectrality of Product Sets with a Perturbed Interval Factor

2025/08/21 by Ramabadran, Aditya, van Vliet, Johannes
#42C99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.15159

Abstract

A set Ω⊂ ℝ^d is said to be spectral if L²(Ω) admits an orthogonal basis of exponentials. While the product of spectral sets is known to be spectral, the converse fails in general. In this paper, we prove that the converse holds when one factor is a perturbation of an interval: if E ⊂ [0,3/2 - ε] and F are bounded sets of measure 1, then E × F is spectral if and only if both E and F are spectral.

Citations

• Spectrality of the product does not imply that the components are spectral
• Orthogonal Fourier Analysis on Domains
• The Fuglede conjecture for convex domains is true in all dimensions
• Spectrality of product domains and Fuglede's conjecture for convex polytopes
• Packing near the tiling density and exponential bases for product domains
• Complex Hadamard matrices and the Spectral Set Conjecture
• Fuglede's conjecture is false in 5 and higher dimensions
• Tiling and spectral properties of near-cubic domains
• Packing, tiling, orthogonality and completeness

Related

• Spectrality of product domains and Fuglede's conjecture for convex polytopes
• Packing near the tiling density and exponential bases for product domains
• Spectrality of the product does not imply that the components are spectral
• Spectrality of Self-Similar Tiles
• On The Rationality Of The Spectrum