NON-SPECTRALITY OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR-ELEMENT DIGIT SETS

In this paper, we consider the non-spectrality of the planar self-affine measures mu(M,) (D) generated by an expanding integer matrix M is an element of M-2(Z) and a four-element digit set D = {(0 0), (alpha(1) alpha(2)), (beta(1) beta(2)), (-alpha(1) - beta(1) -alpha(2) - beta(2))} with alpha(1)beta(2) - alpha(2)beta(1) is not an element of 2Z. We show that L-2(mu(M,) (D)) contains an infinite orthogonal set of exponential functions if and only if det (M) is an element of 2Z. Moreover, if det (M) is an element of 2Z + 1, then there exist at most 4 mutually orthogonal exponential functions in L-2(mu(M,D)), and the number 4 is the best.