The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Let mu(M,D) be the planar self-affine measure determined by an expanding integer matrix M epsilon M-2(Z) and a two-element digit set D subset of Z(2). It has been shown that the spectral or non-spectral problem on mu(M,D) is only related to trace(M) := r(1) and det(M) := r(2). In the case when M is an element of M-2(Z) is an expanding matrix and r(1)(2) = 3r(2), r(2) is an element of 2Z + 1 \ {+/- 1}, the Hilbert space L-2(mu(M,D)) contains at most a finite number of orthogonal exponentials, and mu M, D is a non-spectral measure. The remaining problem in this case is to determine the best upper bound on the cardinality of orthogonal exponentials in the Hilbert space L-2(mu(M,D)). In the present paper, we further the above research to show that there are at most 16 mutually orthogonal exponentials in the corresponding Hilbert space, and the number 16 is the best upper bound. This completes the non-spectrality of self-affine measures in the above case.