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

The self-affine measure μM,DμM,D corresponding to an expanding integer matrixM=[abcd]andD={(00),(10),(01)} is supported on the attractor (or invariant set) of the iterated function system {ϕd(x)=M−1(x+d)}d∈D{ϕd(x)=M−1(x+d)}d∈D. In the present paper we show that if (a+d)2=4(ad−bc)(a+d)2=4(ad−bc) and ad−bcad−bc is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in L2(μM,D)L2(μM,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 μˆM,D, and provides a way of dealing with the non-spectral problem.