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[in Journal: Fractals] AND [Keyword: Zeros] (1)	27 Mar 2025	Run
[Author: Li, X] AND [Keyword: 41A60] (1)	27 Mar 2025	Run
[in Journal: World Scientific Proceedings Series on Computer Engineering and Inform... (1)	27 Mar 2025	Run
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articleNo Access
NON-SPECTRALITY OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR-ELEMENT DIGIT SETS
- JUAN SU,
- YAO LIU, and
- JING-CHENG LIU
Fractals01 Nov 2019
Preview Abstract
In this paper, we consider the non-spectrality of the planar self-affine measures $μ_{M, D}$ generated by an expanding integer matrix $M \in M_{2} (ℤ)$ and a four-element digit set
$D = \{(\begin{matrix} 0 \\ 0 \end{matrix}), (\begin{matrix} α_{1} \\ α_{2} \end{matrix}), (\begin{matrix} β_{1} \\ β_{2} \end{matrix}), (\begin{matrix} - α_{1} - β_{1} \\ - α_{2} - β_{2} \end{matrix})\} with α_{1} β_{2} - α_{2} β_{1} \notin 2 ℤ .$
We show that $L^{2} (μ_{M, D})$ contains an infinite orthogonal set of exponential functions if and only if $det (M) \in 2 ℤ$ . Moreover, if $det (M) \in 2 ℤ + 1$ , then there exist at most $4$ mutually orthogonal exponential functions in $L^{2} (μ_{M, D})$ , and the number $4$ is the best.

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