Non-spectral self-affine measure problem on the plane domain

The self-affine measure mu(M,D) corresponding to an expanding integer matrix [GRAPHICS] is supported on the attractor (or invariant set) of the iterated function system {phi(d)(x) = m(-1)(x +d)}(d epsilon D). In the present paper we show that if (a + d)(2) = 4(ad - bc) and ad - bc is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in L-2(mu(M,D)), and the number 3 is the best. This extends several known results on the non-spectral self-affine measure problem. The proof of such result depends on the characterization of the zero set of the Fourier transform (mu) over cap (M,D), and provides a way of dealing with the non-spectral problem. (C) 2010 Elsevier Inc. All rights reserved.