Abstract
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
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 \(\eta =\max \{r: 3^r| (\alpha _1\alpha _4-\alpha _2\alpha _3)\}\).
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Acknowledgements
The author would like to thank the anonymous referees for their valuable suggestions and comments leading to the improvement of our manuscript. The research is supported in part by the NNSF of China (No. 11831007) and a project supported by Scientific Research Fund of Hunan Provincial Education Department (No. 19C0579).
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Communicated by Baruch Solel.
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Wang, ZM. Non-spectral problem for the planar self-affine measures with decomposable digit sets. Ann. Funct. Anal. 11, 1287–1297 (2020). https://doi.org/10.1007/s43034-020-00086-6
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DOI: https://doi.org/10.1007/s43034-020-00086-6