In this paper, we consider the non-spectral problem for the planar self-affine measures $$\mu _{M,D}$$ generated by an expanding integer matrix $$M\in M_2({\mathbb {Z}})$$ and a finite digit set $$\begin{aligned} D=\{(0,0)^t, (\alpha _1,\alpha _2)^t, (\alpha _3, \alpha _4)^t \} \oplus k(\alpha _1\alpha _4-\alpha _2\alpha _3){\mathfrak {D}}, \end{aligned}$$ where $$k\in {\mathbb {Z}}\backslash \{0\}, \alpha _i\in {\mathbb {Z}}\;(1\le i\le 4)$$ and $${\mathfrak {D}\subset \mathbb{R}^2}$$ is a finite integer digit set. Let $$Z(m_D)=\{x\in {\mathbb {R}}^2: \sum _{d\in D} e^{2\pi i \langle d, x\rangle }=0\}$$ and $$\mathring{E_3^2}:=\frac{1}{3}\{(l_1, l_2)^t: l_1, l_2\in \mathbb{N},0\le l_1, l_2\le 2\}\backslash \{0\}$$ . We prove that if $$\alpha _1\alpha _4-\alpha _2\alpha _3\ne 0$$ , $$Z(m_{{\mathfrak {D}}})\subset \mathring{E_{3 }^2} \pmod {{\mathbb {Z}}^2}$$ and $$\gcd (\det (M), 3)=1$$ , then there exist at most $$\max \{17,9^{\eta }+8\}$$ mutually orthogonal exponential functions in $$L^2(\mu _{M,D})$$ , where $$\eta =\max \{r: 3^r| (\alpha _1\alpha _4-\alpha _2\alpha _3)\}$$ .

中文翻译：

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具有可分解数字集的平面自仿射测度的非谱问题

在本文中，我们考虑了由扩展整数矩阵 $$M\in M_2({\mathbb {Z}} )$$ 和有限数字集 $$\begin{aligned} D=\{(0,0)^t, (\alpha _1,\alpha _2)^t, (\alpha _3, \alpha _4)^t \} \oplus k(\alpha _1\alpha _4-\alpha _2\alpha _3){\mathfrak {D}}, \end{aligned}$$ where $$k\in {\mathbb {Z}}\backslash \{0\}, \alpha _i\in {\mathbb {Z}}\;(1\le i\le 4)$$ 和 $${\mathfrak {D}\subset \mathbb{R}^2} $$ 是一个有限整数数字集。让 $$Z(m_D)=\{x\in {\mathbb {R}}^2: \sum _{d\in D} e^{2\pi i \langle d, x\rangle }=0\ }$$ 和 $$\mathring{E_3^2}:=\frac{1}{3}\{(l_1, l_2)^t: l_1, l_2\in \mathbb{N},0\le l_1, l_2 \le 2\}\反斜杠 \{0\}$$ 。我们证明如果 $$\alpha _1\alpha _4-\alpha _2\alpha _3\ne 0$$ ，$$Z(m_{{\mathfrak {D}}})\subset \mathring{E_{3 }^2} \pmod {{\mathbb {Z}}^2}$$ 和 $$\gcd (\det (M), 3)=1$$ ，则在 $$L^2(\mu _ {M,D})$$ ，其中 $$\eta =\max \{r: 3^r| (\alpha _1\alpha _4-\alpha _2\alpha _3)\}$$ 。