Non-spectral problem for a class of planar self-affine measures

The self-affine measure μM,D corresponding to an expanding matrix M ∈ Mn(R) and a finite subset D ⊂ Rn is supported on the attractor (or invariant set) of the iterated function system {φd (x) = M −1(x+d)} d∈D. The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have 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 if a, b, c ∈ Z, |a| > 1, |c| > 1 and ac ∈ Z \ (3Z), M = a b 0 c and D = 0 0 , 1 0 , 0 1 , then there exist at most 3 mutually orthogonal exponentials in L2(μM,D), and the number 3 is the best. This extends several known conclusions. The proof of such result depends on the characterization of the zero set of the Fourier transform ˆμM,D, and provides a way of dealing with the non-spectral problem. © 2008 Elsevier Inc. All rights reserved.