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

TL;DR

This paper investigates the maximum number of mutually orthogonal exponential functions in the space of a specific planar self-affine measure, providing explicit bounds based on the properties of the defining matrix and digit set.

Contribution

It establishes a criterion linking the determinant of the matrix to the finiteness of orthogonal exponentials and determines the exact maximal cardinality when the determinant condition is met.

Findings

Orthogonal exponential functions are finite if det(M) is not divisible by 3.

The exact maximal number of orthogonal exponentials is characterized.

Provides explicit formulas for the maximal cardinality based on matrix properties.

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.