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arXiv · 1805.09941

The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets

Ming-Liang Chen·Jing-Cheng Liu
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Abstract

In this paper, we consider the planar self-affine measures $\mu_{M,D}$ generated by an expanding matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $ D=\left\{ {\left( {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} \alpha_1\\ \alpha_2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} \beta_1\\ \beta_2 \end{array}} \right)} \right\} $ with $\alpha_1\beta_2-\alpha_2\beta_1\neq0$. We show that if $\det(M)\notin 3\mathbb{Z}$, then the mutually orthogonal exponential functions in $L^2(\mu_{M,D})$ is finite, and the exact maximal cardinality is given.

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Ming-Liang Chen, Jing-Cheng Liu. 2018-05-25. The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets. https://arxiv.org/abs/1805.09941

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