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

Abstract

We study spectral properties of the self-affine measure ${\mu }_{M,\mathcal{D}}$ generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a consecutive collinear digit set $\mathcal{D}=\{0,1,\dots ,q-1\}v$ 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. Moreover, for some special cases, we can obtain a necessary and sufficient condition on 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}).

14 pages, published by JFA

References in corpus (2)

• Complex Hadamard matrices and the Spectral Set Conjecture
• Hadamard triples generate self-affine spectral measures

Cited by in corpus (2)

• The spectrality of self-affine measure under the similarity transformation of $G{L}_n(p)$
• Spectrality of generalized Sierpinski-type self-affine measures