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

Abstract

In a previous work by Łaba and Wang, it was proved that whenever there is a Hadamard triple (N,D,L), then the associated one-dimensional self-similar measure μN,D generated by maps N−1(x+d) with d∈D, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples (N,Ak,Lk) by putting each triple into different scales of N. Our main result is to prove that the associated self-similar measure μN,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 ZN obey the Coven-Meyerowitz (T1), (T2) tiling condition. Moreover, we show that all self-similar tiles with N=pαq 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.