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

Abstract

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 dIn 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 (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 transformμ M,D , and provides a way of dealing with the non-spectral problem.