Spectral property of self-affine measures on ${\mathbb R}^n$

Jing-Cheng Liu, Jun Jason Luo

Published 2016-03-19Version 1

We study the spectrality of the self-affine measure $\mu_{M,\mathcal {D}}$ generated by expanding integer matrices $M\in M_n(\mathbb{Z})$ and consecutive collinear digit sets $\mathcal {D}=\{0,1,\dots,q-1\}v$ where $v\in \mathbb{Z}^n\setminus\{0\}$ and $q\ge 2$ is an integer. Some sufficient conditions for $\mu_{M,\mathcal {D}}$ to be a spectral measure or to have infinitely many orthogonal exponentials are given. In particular, when $v$ is an eigenvector of $M$ or the characteristic polynomial of $M$ is of the simple form $f(x)=x^n+c$, we obtain a necessary and sufficient condition of the spectrality of $\mu_{M,\mathcal {D}}$. Our study generalizes the one dimensional results proved by Dai, {\it et al.} (\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}).