Non-spectrality of planar self-affine measures with three-elements digit set

The self-affine measure μM,DμM,D associated with an affine iterated function system {ϕd(x)=M−1(x+d)}d∈D{ϕd(x)=M−1(x+d)}d∈D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM,DμM,D have been received much attention in recent years. One of the non-spectral problem on μM,DμM,D is to estimate the number of orthogonal exponentials in L2(μM,D)L2(μM,D) and to find them. In the present paper we show that for an expanding integer matrix M∈M2(Z)M∈M2(Z) and the three-elements digit set D given byM=[abdc]andD={(00),(10),(01)}, if ac−bd∉3Zac−bd∉3Z, then there exist at most 3 mutually orthogonal exponentials in L2(μM,D)L2(μM,D), and the number 3 is the best. This confirms the three-elements digit set conjecture on the non-spectrality of self-affine measures in the plane