Spectrality of Product Sets with a Perturbed Interval Factor

TL;DR

This paper investigates the spectrality of product sets, proving that when one factor is a perturbed interval, the product's spectrality is equivalent to both factors being spectral, extending known results in harmonic analysis.

Contribution

It establishes a new equivalence condition for spectrality of product sets involving a perturbed interval factor, filling a gap in the understanding of spectral sets.

Findings

Product of spectral sets is spectral, but the converse is not always true.

When one factor is a perturbed interval, the product is spectral iff both factors are spectral.

The result applies to bounded sets of measure 1 within specific interval bounds.

Abstract

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if $L^2(\Omega)$ 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 $E \subset [0,3/2 - \epsilon]$ and $F$ are bounded sets of measure $1$, then $E \times F$ is spectral if and only if both $E$ and $F$ are spectral.