  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 $\Omega$.
