Product-form Hadamard triples and its spectral self-similar measures

In a previous work by Łaba and Wang, it was proved that whenever there is a Hadamard triple $(N,\mathcal{D},\mathcal{L})$, then the associated one-dimensional self-similar measure ${\mu }_{N,\mathcal{D}}$ generated by maps $N^{-1}(x+d)$ with $d\in \mathcal{D}$, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples $(N,{\mathcal{A}}_k,{\mathcal{L}}_k)$ by putting each triple into different scales of $N$. Our main result is to prove that the associated self-similar measure ${\mu }_{N,\mathcal{D}}$ is a spectral measure. This result allows us to show that product-form self-similar tiles are spectral sets as long as the tiles in the group ${\mathbb{Z}}_N$ obey the Coven-Meyerowitz $(T1)$, $(T2)$ tiling condition. Moreover, we show that all self-similar tiles with $N=p^{\alpha }q$ are spectral sets, answering a question by Fu, He and Lau in 2015. Finally, our results allow us to offer new singular spectral measures not generated by a single Hadamard triple. Such new examples allow us to classify all spectral self-similar measures generated by four equi-contraction maps, which will appear in a forthcoming paper.