The cardinality of orthogonal exponentials of planar self-affine measures with three-element digit sets (1805.09941v1)

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.