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

Abstract

In a previous work by {\L}aba 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.