Spectral Property of Self-Affine Measures on $\pmb {\mathbb {R}^n}$

Abstract

For any integer $m>1$, let $D\subset {{\mathbb {Z}}}^n$ be a finite digit set such that ${{\mathcal {Z}}}(m_D)=\cup _{i=1}^k{{\mathcal {Z}}}_i$ for some finite integer k, $({{\mathcal {Z}}}_i-{{\mathcal {Z}}}_i)\setminus \mathbb {Z}^n\subset {{\mathcal {Z}}}_i\subset (m^{-1}{{\mathbb {Z}}}\setminus {{\mathbb {Z}}})^n$ and ${{\mathcal {Z}}}_i\not \subset ({m'}^{-1}{{\mathbb {Z}}}\setminus {{\mathbb {Z}}})^n$ for all $0<m'<m$, where ${{\mathcal {Z}}}(m_D)=\big \{x:\sum _{d\in D}e^{2\pi i\left\langle d,x \right\rangle }=0\big \}$. Let $M={\hbox {diag}}[b_1,\ldots ,b_n]$ be a real expansive diagonal matrix and $\mu _{M, D}$ be the self-affine measure on ${{\mathbb {R}}}^n$ defined by $\mu _{M,D}(\cdot )=\frac{1}{|D|}\sum _{d\in D}\mu _{M, D}(M(\cdot )-d)$. In this paper, we first give the sufficient and necessary condition for $L^2(\mu _{M,D})$ to contain an infinite orthogonal exponentials for any integer $m>1$. Furthermore, we show that, if m is a prime, $\mu _{M, D}$ is a spectral measure if and only if $m|b_i$, $i=1,2,\ldots ,n$. This extends known results in [5, 6, 28].