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 μ M,D generated by an expanding matrix M∈M 2 (Z)and an integer digit set D={(00),(α 1 α 2 ),(β 1 β 2 )} with α 1 β 2 −α 2 β 1 ≠0. We show that if det⁡(M)∉3Z, then the mutually orthogonal exponential functions in L 2 (μ M,D )is finite, and the exact maximal cardinality is given.