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

The self-affine measure μ M , D corresponding to an expanding integer matrix M = [ a b c d ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) } is supported on the attractor (or invariant set) of the iterated function system { ϕ d ( x ) = M − 1 ( x + d ) } d ∈ D . In the present paper we show that if ( a + d ) 2 = 4 ( a d − b c ) and a d − b c is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in L 2 ( μ 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 μ ˆ M , D , and provides a way of dealing with the non-spectral problem.