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

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}} α_1\\ α_2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} β_1\\ β_2 \end{array}} \right)} \right\} $ with ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\ne 0$. 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.