Pro duct-form Hadamard triples and its spectral self-similar measures

摘要

In a previous work by Laba and Wang, it was proved that whenever there is a Hadamard triple (N, D, L), then the associated one-dimensional self-similar measure mu N,D generated by maps N-1(x + d) with d is an element of D, is a spectral measure. In this paper, we introduce pro duct-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 mu N,D is a spectral measure. This result allows us to show that pro duct-form self-similar tiles are spectral sets as long as the tiles in the group ZN obey the Coven-Meyerowitz (T 1), (T 2) 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.

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