2022/09/12 by An, Li-Xiang, Lai, Chun-Kit · 2 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2209.05616
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 μ_N,\mathcal D generated by maps N-1(x+d) with d∈\mathcal D, is a spectral measure. In this paper, we introduce product-form digit sets for finitely many Hadamard triples (N, \mathcal Ak, \mathcal Lk) by putting each triple into different scales of N. Our main result is to prove that the associated self-similar measure μ_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 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.