Spectrality of factors of product spectral measures

We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting.

Furthermore, we establish a new class of spectral unions of intervals for which the product spectral set question has a positive answer.

More precisely, if $A$ is a subset of the natural numbers such that $A\oplus B = \{0,1,\cdots, N-1\}$ for some $B\subset \mathbb N$ and $N>1$ then the product measure $\mathcal{L}|_{A+[0,1]}\times \nu$ is a spectral measure (that may be singular) if and only if $\nu$ is a spectral measure.