WITHDRAWN: Spectral and non-spectral property of self-affine measures on ℝn

Abstract In this paper, we study twofold questions regarding the spectrality and non-spectrality of the self-affine measure $\mu_{M,\mathcal{D}}$. Our result characterizes a necessary and sufficient condition on the spectrality of $\mu_{M,\mathcal{D}}$ generated by a generalized symmetric expanding integer matrix $M\in M_{n}(\mathbb{Z})$ and a non-consecutive collinear digit set $\mathcal{D}$. For the lower triangular expanding integer matrix $M\in M_{n}(\mathbb{Z} )$ and $\mathcal{D}\subset \mathbb{Z}^{n}$, let $\mathcal{Z}_{\mathcal{D}}^{n}:=\{x\in [0,1)^{n} : m_{\mathcal{D}}(x)=0\}$ and $N_{m}^{n}=\{(\bm{r}, \bm{q})\in \mathbb{R}^{n}:\bm{r}:=(r_1,r_2,\ldots,r_{n-m})^{t}\in [0,1)^{n-m}, \bm{q}\in \mathbb{Q}^{m}\setminus \{\bm{0}\} \}$. Denote $p_{m,\mathcal{D}}$ as the positive integers which are determined by $m$ and $\mathcal{D}$ for any integer $m\geq 1$. We show that if $\gcd(\det(M), p_{m,\mathcal{D}})=1$ for $m\geq 1$ and $\emptyset\neq \mathcal{Z}_{\mathcal{D}}^{n}\subset N_{m}^{n}$, then there are at most $(p_{m,\mathcal{D}})^{m}$ mutually orthogonal exponential functions in $L^{2}(\mu_{M,D})$.