There are Four-Element Orthogonal Exponentials of Planar Self-affine Measures with Two Digits

Let mu(M,D) be the self-affine measure associated with an expanding integer matrix M = (p 0 , 0 q) and D = {(0 0), (1 1)}, where vertical bar p vertical bar and vertical bar q vertical bar are distinct odd bigger than 1. Such a measure is the simplest and the most important case in the study of the spectral property of self-affine measures with two-elements digit sets, which is an open problem up to now. In this paper, we first construct two classes of 4-element orthogonal exponentials in the corresponding Hilbert space L-2(mu(M,D)). Moreover, we prove that, under certain conditions, the constructed 4-element orthogonal exponentials is maximal.