Spectrality of Product Sets with a Perturbed Interval Factor

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