Packing near the tiling density and exponential bases for product domains

A set $\Omega$ in a locally compact abelian group is called spectral if $L^2(\Omega)$ has an orthogonal basis of group characters. An important problem, connected with the so-called Spectral Set Conjecture (saying that $\Omega$ is spectral if and only if a collection of translates of $\Omega$ can partition the group), is the question of whether the spectrality of a product set $\Omega = A \times B$, in a product group, implies the spectrality of the factors $A$ and $B$. Recently Greenfeld and Lev proved that if $I$ is an interval and $\Omega \subseteq {\mathbb R}^d$ then the spectrality of $I \times \Omega$ implies the spectrality of $\Omega$. We give a different proof of this fact by first proving a result about packings of high density implying the existence of tilings by translates of a function. This allows us to improve the result to a wider collection of product sets than those dealt with by Greenfeld and Lev. For instance when $A$ is a union of two intervals in ${\mathbb R}$ then we show that the spectrality of $A \times \Omega$ implies the spectrality of both $A$ and $\Omega$.