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

The self-affine measure μM,D associated with an affine iterated function system {φd (x) = M−1(x + d)}d∈D is uniquely determined. The problems of determining the spectrality or non-spectrality of a measure μM,D have been received much attention in recent years. One of the non-spectral problem on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D) and to find them. In the present paper we show that for an expanding integer matrix M ∈ M2(Z) and the three-elements digit set D given by M = a b d c and D = 0 0 , 1 0 , 0 1 , if ac−bd /∈ 3Z, then there exist at most 3 mutually orthogonal exponentials in 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 © 2008 Elsevier Inc. All rights reserved