On the spectrality of self-affine measures with four digits on R2

In this paper, we study the spectral property of the self-affine measure mu(R,D) generated by an expanding real matrix R = diag (b, b) and the four-element digit set D = { (0 0), (1 0), (0 1), (-1 -1) }. We show that mu(R,D) is a spectral measure, i.e. there exists a discrete set Lambda subset of R-2 such that the collection of exponential functions {e(-2 pi iota <lambda,x >) : lambda is an element of Lambda} forms an orthonormal basis for L-2(mu), if and only if b = 2k for some k is an element of N. A similar characterization for Bernoulli convolution is provided by Dai [X. -R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681-1693], over which b = 2k. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of mu(R,D) by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187-208]. We also extend these results to the more general self-affine measures.