ORTHOGONAL EXPONENTIAL FUNCTIONS OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

For the self-affine measure mu(M,D) generated by an expanding matrix M is an element of M-2(Z) and an integer digit set D={((0)(0)), ((alpha 1)(alpha 2)), ((beta)(1)(beta 2)), ( [GRAPHICS] } with alpha(1)beta(2) - alpha(2)/beta(1)not equal 0, Su et al. proved that if alpha(1)beta(2) - alpha(2)/beta(1)is not an element of 2Z, then L-2(mu(M,D)) contains an infinite orthogonal set of exponential functions if and only if det(M) is an element of 2Z [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, &adds (2019), https:// doi .org/10.1142/S0218348X19501159] . in this paper, we show that the above conclusion also holds for alpha(1)beta(2) - alpha(2)/beta(1) is an element of 2Z. So, a complete characterization of L-2(mu(M,D)) containing an infinite orthogonal set of exponential functions is given.