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A set $\Omega$ in a locally compact abelian group is called spectral if$L^2(\Omega)$ has an orthogonal basis of group characters. An importantproblem, connected with the so-called Spectral Set Conjecture (saying that$\Omega$ is spectral if and only if a collection of translates of $\Omega$ canpartition the group), is the question of whether the spectrality of a productset $\Omega = A \times B$, in a product group, implies the spectrality of thefactors $A$ and $B$. Recently Greenfeld and Lev proved that if $I$ is aninterval and $\Omega \subseteq {\mathbb R}^d$ then the spectrality of $I \times\Omega$ implies the spectrality of $\Omega$. We give a different proof of thisfact by first proving a result about packings of high density implying theexistence of tilings by translates of a function. This allows us to improve theresult to a wider collection of product sets than those dealt with by Greenfeldand 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 spectralityof both $A$ and $\Omega$.

Packing near the tiling density and exponential bases for product domains